In this paper, we are devoted to establishing the point-wise decay estimates for solution to the 5D defocusing energy-critical Hartree equation with an initial data in H2(R5)∩L1(R5)${H}^{2}left({mathbb{R}}^{5}right)cap {L}^{1}left({mathbb{R}}^{5}right)$. We show that the nonlinear solution has the same time decay rate as the linear one. The main new ingredient is that we used the theories of Lorentz spaces to overcome the low power of nonlinearity.
本文致力于建立初始数据为 H 2 ( R 5 ) ∩ L 1 ( R 5 ) ${H}^{2}left({mathbb{R}}^{5}right)cap {L}^{1}left({mathbb{R}}^{5}right)$ 的 5D 失焦能量临界哈特里方程解的随点衰减估计。我们证明，非线性解与线性解具有相同的时间衰减率。主要的新成分是我们利用洛伦兹空间理论克服了非线性的低功率问题。
{"title":"Decay estimates for defocusing energy-critical Hartree equation","authors":"Miao Chen, Hua Wang, Xiaohua Yao","doi":"10.1515/ans-2023-0138","DOIUrl":"https://doi.org/10.1515/ans-2023-0138","url":null,"abstract":"In this paper, we are devoted to establishing the point-wise decay estimates for solution to the 5D defocusing energy-critical Hartree equation with an initial data in <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:msup> <m:mrow> <m:mi>H</m:mi> </m:mrow> <m:mrow> <m:mn>2</m:mn> </m:mrow> </m:msup> <m:mrow> <m:mo stretchy=\"false\">(</m:mo> <m:mrow> <m:msup> <m:mrow> <m:mi mathvariant=\"double-struck\">R</m:mi> </m:mrow> <m:mrow> <m:mn>5</m:mn> </m:mrow> </m:msup> </m:mrow> <m:mo stretchy=\"false\">)</m:mo> </m:mrow> <m:mo>∩</m:mo> <m:msup> <m:mrow> <m:mi>L</m:mi> </m:mrow> <m:mrow> <m:mn>1</m:mn> </m:mrow> </m:msup> <m:mrow> <m:mo stretchy=\"false\">(</m:mo> <m:mrow> <m:msup> <m:mrow> <m:mi mathvariant=\"double-struck\">R</m:mi> </m:mrow> <m:mrow> <m:mn>5</m:mn> </m:mrow> </m:msup> </m:mrow> <m:mo stretchy=\"false\">)</m:mo> </m:mrow> </m:math> <jats:tex-math>${H}^{2}left({mathbb{R}}^{5}right)cap {L}^{1}left({mathbb{R}}^{5}right)$</jats:tex-math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_ans-2023-0138_ineq_001.png\"/> </jats:alternatives> </jats:inline-formula>. We show that the nonlinear solution has the same time decay rate as the linear one. The main new ingredient is that we used the theories of Lorentz spaces to overcome the low power of nonlinearity.","PeriodicalId":7191,"journal":{"name":"Advanced Nonlinear Studies","volume":"21 1","pages":""},"PeriodicalIF":1.8,"publicationDate":"2024-05-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140840659","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
本文的目的有三个方面。首先，我们建立了限制较少的奇异特鲁丁格-莫泽不等式： (0.1) sup u ∈ H 1 ( R 2 ) , ∫ R 2 ( |∇ u | 2 + V ( x ) u 2 ) d x ≤ 1 ∫ R 2 e 4 π 1 - β 2 u 2 - 1 | x | β d x <；+ ∞ , $$underset{uin {H}^{1}({mathbb{R}}^{2})、underset{{mathbb{R}}^{2}}{int }(vert nabla u{vert }^{2}+V(x){u}^{2})mathrm{d}xle 1}{mathrm{sup}}underset{{mathbb{R}}^{2}}{int }frac{e}^{4pi left(1-).tfrac{beta }{2}right){u}^{2}}-1}{vert x{vert }^{beta }}mathrm{d}x<；+infty ,$$ 其中 0 < β < 2 $0< beta < 2$ , V ( x ) ≥ 0 $V(x)ge 0$ 并且可能在 R 2 ${mathbb{R}}^{2}$ 中的开集上消失。其次，我们考虑在 R 2 ${mathbb{R}}^{2}$ 中存在以下具有临界指数增长的薛定谔方程的基态： (0.2) - Δ u + γ u = f ( u ) | x | β , $${-}{Delta }u+gamma u=frac{f(u)}{vert x{vert }^{beta }},$$其中非线性 f $f$ 具有临界指数增长。为了克服紧凑性的不足，我们开发了一种基于最小能量阈值的方法，该方法是嵌入定理（C. Zhang and L. Chen, "Concentration-compactness principle of singular Trudinger-Moser inequalities in R n ${mathbb{R}}^{n}$ and n $n$ -Laplace equations," Adv. Nonlinear Stud.3, pp. 567-585, 2018）和 Nehari 流形，从而得到基态的存在。此外，作为不等式（0.1）的应用，我们还证明了涉及 R 2 ${mathbb{R}}^{2}$ 中退化势的下列方程的基态存在性： (0.3) - Δ u + V ( x ) u = f ( u ) | x | β . $${-}{Delta }u+V(x)u=frac{f(u)}{vert xvert }^{beta }}.$$$
In a smoothly bounded convex domain <jats:inline-formula> <jats:alternatives> <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <m:mi mathvariant="normal">Ω</m:mi> <m:mo>⊂</m:mo> <m:msup> <m:mrow> <m:mi mathvariant="double-struck">R</m:mi> </m:mrow> <m:mrow> <m:mi>n</m:mi> </m:mrow> </m:msup> </m:math> <jats:tex-math>${Omega}subset {mathbb{R}}^{n}$</jats:tex-math> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_ans-2023-0131_ineq_001.png"/> </jats:alternatives> </jats:inline-formula> with <jats:italic>n</jats:italic> ≥ 1, a no-flux initial-boundary value problem for<jats:disp-formula> <jats:alternatives> <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" display="block" overflow="scroll"> <m:mfenced close="" open="{"> <m:mrow> <m:mtable> <m:mtr> <m:mtd columnalign="left"> <m:msub> <m:mrow> <m:mi>u</m:mi> </m:mrow> <m:mrow> <m:mi>t</m:mi> </m:mrow> </m:msub> <m:mo>=</m:mo> <m:mi mathvariant="normal">Δ</m:mi> <m:mfenced close=")" open="("> <m:mrow> <m:mi>u</m:mi> <m:mi>ϕ</m:mi> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mrow> <m:mi>v</m:mi> </m:mrow> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:mfenced> <m:mo>,</m:mo> <m:mspace width="1em"/> </m:mtd> </m:mtr> <m:mtr> <m:mtd columnalign="left"> <m:msub> <m:mrow> <m:mi>v</m:mi> </m:mrow> <m:mrow> <m:mi>t</m:mi> </m:mrow> </m:msub> <m:mo>=</m:mo> <m:mi mathvariant="normal">Δ</m:mi> <m:mi>v</m:mi> <m:mo>−</m:mo> <m:mi>u</m:mi> <m:mi>v</m:mi> <m:mo>,</m:mo> <m:mspace width="1em"/> </m:mtd> </m:mtr> </m:mtable> </m:mrow> </m:mfenced> </m:math> <jats:tex-math>$$begin{cases}_{t}={Delta}left(uphi left(vright)right),quad hfill {v}_{t}={Delta}v-uv,quad hfill end{cases}$$</jats:tex-math> <jats:graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_ans-2023-0131_eq_999.png"/> </jats:alternatives> </jats:disp-formula>is considered under the assumption that near the origin, the function <jats:italic>ϕ</jats:italic> suitably generalizes the prototype given by<jats:disp-formula> <jats:alternatives> <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" display="block" overflow="scroll"> <m:mi>ϕ</m:mi> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mrow> <m:mi>ξ</m:mi> </m:mrow> <m:mo stretchy="false">)</m:mo> </m:mrow> <m:mo>=</m:mo> <m:msup> <m:mrow> <m:mi>ξ</m:mi> </m:mrow> <m:mrow> <m:mi>α</m:mi> </m:mrow> </m:msup> <m:mo>,</m:mo> <m:mspace width="2em"/> <m:mi>ξ</m:mi> <m:mo>∈</m:mo> <m:mrow> <m:mo stretchy="false">[</m:mo> <m:mrow> <m:mn>0</m:mn> <m:mo>,</m:mo> <m:msub> <m:mrow> <m:mi>ξ</m:mi> </m:mrow> <m:mrow> <m:mn>0</m:mn> </m:mrow> </m:msub> </m:mrow> <m:mo stretchy="false">]</m:mo> </m:mrow> <m:mo>.</m:mo> </m:math> <jats:tex-math>$$phi left(xi right)={xi }^{alpha },qquad xi in left[0,{xi }_{0}right].$$</jats:tex-math> <jats:graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_ans-2023-0131_eq_998.png"/> </jats:alternatives> </jats:disp-formula>By means of separate approaches, it is shown that in bo
在一个 n ≥ 1 的平滑有界凸域 Ω ⊂ R n ${Omega}subset {mathbb{R}}^{n}$ 中，对 u t = Δ u ϕ ( v ) , v t = Δ v - u v , $$begin{cases}_{t}={Delta}left(uphi left(vright)right) 的无流初始边界值问题进行了研究、quad hfill {v}_{t}={Delta}v-uv,quad hfill end{cases}$$ 是在这样的假设下考虑的，即在原点附近，函数j适当地概括了原型：j ( ξ ) = ξ α , ξ∈ [ 0 , ξ 0 ] 。 $$phi left(xi right)={xi }^{α },qquad xi in left[0,{xi }_{0}right].$$ 通过不同的方法表明，在 α∈ (0, 1) 和 α∈ [1, 2] 两种情况下，都存在一些全局弱解，这些弱解满足 C ( T ) ≔ ess sup t∈ ( 0 , T ) ∫ Ω u ( ⋅ , t ) ln u ( ⋅ , t ) < ∞ for all T > 0 , $$Cleft(Tright){：=}underset{tin left(0,Tright)}{text{ess,sup}}{int}_{{Omega}}uleft(cdot ,tright)mathrm{ln}uleft(cdot ,tright){<；}infty qquad text{for,all,}T{ >}0,$$ with sup T>0 C(T) < ∞ if α∈ [1, 2].
{"title":"A degenerate migration-consumption model in domains of arbitrary dimension","authors":"Michael Winkler","doi":"10.1515/ans-2023-0131","DOIUrl":"https://doi.org/10.1515/ans-2023-0131","url":null,"abstract":"In a smoothly bounded convex domain <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\" overflow=\"scroll\"> <m:mi mathvariant=\"normal\">Ω</m:mi> <m:mo>⊂</m:mo> <m:msup> <m:mrow> <m:mi mathvariant=\"double-struck\">R</m:mi> </m:mrow> <m:mrow> <m:mi>n</m:mi> </m:mrow> </m:msup> </m:math> <jats:tex-math>${Omega}subset {mathbb{R}}^{n}$</jats:tex-math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_ans-2023-0131_ineq_001.png\"/> </jats:alternatives> </jats:inline-formula> with <jats:italic>n</jats:italic> ≥ 1, a no-flux initial-boundary value problem for<jats:disp-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\" display=\"block\" overflow=\"scroll\"> <m:mfenced close=\"\" open=\"{\"> <m:mrow> <m:mtable> <m:mtr> <m:mtd columnalign=\"left\"> <m:msub> <m:mrow> <m:mi>u</m:mi> </m:mrow> <m:mrow> <m:mi>t</m:mi> </m:mrow> </m:msub> <m:mo>=</m:mo> <m:mi mathvariant=\"normal\">Δ</m:mi> <m:mfenced close=\")\" open=\"(\"> <m:mrow> <m:mi>u</m:mi> <m:mi>ϕ</m:mi> <m:mrow> <m:mo stretchy=\"false\">(</m:mo> <m:mrow> <m:mi>v</m:mi> </m:mrow> <m:mo stretchy=\"false\">)</m:mo> </m:mrow> </m:mrow> </m:mfenced> <m:mo>,</m:mo> <m:mspace width=\"1em\"/> </m:mtd> </m:mtr> <m:mtr> <m:mtd columnalign=\"left\"> <m:msub> <m:mrow> <m:mi>v</m:mi> </m:mrow> <m:mrow> <m:mi>t</m:mi> </m:mrow> </m:msub> <m:mo>=</m:mo> <m:mi mathvariant=\"normal\">Δ</m:mi> <m:mi>v</m:mi> <m:mo>−</m:mo> <m:mi>u</m:mi> <m:mi>v</m:mi> <m:mo>,</m:mo> <m:mspace width=\"1em\"/> </m:mtd> </m:mtr> </m:mtable> </m:mrow> </m:mfenced> </m:math> <jats:tex-math>$$begin{cases}_{t}={Delta}left(uphi left(vright)right),quad hfill {v}_{t}={Delta}v-uv,quad hfill end{cases}$$</jats:tex-math> <jats:graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_ans-2023-0131_eq_999.png\"/> </jats:alternatives> </jats:disp-formula>is considered under the assumption that near the origin, the function <jats:italic>ϕ</jats:italic> suitably generalizes the prototype given by<jats:disp-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\" display=\"block\" overflow=\"scroll\"> <m:mi>ϕ</m:mi> <m:mrow> <m:mo stretchy=\"false\">(</m:mo> <m:mrow> <m:mi>ξ</m:mi> </m:mrow> <m:mo stretchy=\"false\">)</m:mo> </m:mrow> <m:mo>=</m:mo> <m:msup> <m:mrow> <m:mi>ξ</m:mi> </m:mrow> <m:mrow> <m:mi>α</m:mi> </m:mrow> </m:msup> <m:mo>,</m:mo> <m:mspace width=\"2em\"/> <m:mi>ξ</m:mi> <m:mo>∈</m:mo> <m:mrow> <m:mo stretchy=\"false\">[</m:mo> <m:mrow> <m:mn>0</m:mn> <m:mo>,</m:mo> <m:msub> <m:mrow> <m:mi>ξ</m:mi> </m:mrow> <m:mrow> <m:mn>0</m:mn> </m:mrow> </m:msub> </m:mrow> <m:mo stretchy=\"false\">]</m:mo> </m:mrow> <m:mo>.</m:mo> </m:math> <jats:tex-math>$$phi left(xi right)={xi }^{alpha },qquad xi in left[0,{xi }_{0}right].$$</jats:tex-math> <jats:graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_ans-2023-0131_eq_998.png\"/> </jats:alternatives> </jats:disp-formula>By means of separate approaches, it is shown that in bo","PeriodicalId":7191,"journal":{"name":"Advanced Nonlinear Studies","volume":"64 1","pages":""},"PeriodicalIF":1.8,"publicationDate":"2024-04-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140840862","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this paper, we study the problem of prescribing Q-Curvature on higher dimensional standard spheres. The problem consists in finding the right assumptions on a function K so that it is the Q-Curvature of a metric conformal to the standard one on the sphere. Using some pinching condition, we track the change in topology that occurs when crossing a critical level (or a virtually critical level if it is a critical point at infinity) and then compute a certain Euler-Poincaré index which allows us to prove the existence of many solutions. The locations of the levels sets of these solutions are determined in a very precise manner. These type of multiplicity results are new and are proved without any assumption of symmetry or periodicity on the function K.
在本文中，我们研究了在高维标准球面上规定 Q-Curvature 的问题。问题在于找到函数 K 的正确假设，使其成为与球面上标准度量一致的 Q曲率。利用一些捏合条件，我们可以跟踪跨越临界水平（如果是无穷远处的临界点，则为实际上的临界水平）时发生的拓扑变化，然后计算一定的欧拉-平卡指数，从而证明许多解的存在。这些解的水平集位置是以非常精确的方式确定的。这类多重性结果是全新的，无需假设函数 K 的对称性或周期性即可证明。
{"title":"New multiplicity results in prescribing Q-curvature on standard spheres","authors":"Mohamed Ben Ayed, Khalil El Mehdi","doi":"10.1515/ans-2023-0135","DOIUrl":"https://doi.org/10.1515/ans-2023-0135","url":null,"abstract":"In this paper, we study the problem of prescribing <jats:italic>Q</jats:italic>-Curvature on higher dimensional standard spheres. The problem consists in finding the right assumptions on a function <jats:italic>K</jats:italic> so that it is the <jats:italic>Q</jats:italic>-Curvature of a metric conformal to the standard one on the sphere. Using some pinching condition, we track the change in topology that occurs when crossing a critical level (or a virtually critical level if it is a critical point at infinity) and then compute a certain Euler-Poincaré index which allows us to prove the existence of many solutions. The locations of the levels sets of these solutions are determined in a very precise manner. These type of multiplicity results are new and are proved without any assumption of symmetry or periodicity on the function <jats:italic>K</jats:italic>.","PeriodicalId":7191,"journal":{"name":"Advanced Nonlinear Studies","volume":"51 1","pages":""},"PeriodicalIF":1.8,"publicationDate":"2024-04-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140623905","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Francisco Ortegón Gallego, Mohamed Rhoudaf, Hajar Talbi
In this paper, we analyze the following nonlinear elliptic problem A(u)=ρ(u)|∇φ|2 in Ω,div(ρ(u)∇φ)=0 in Ω,u=0 on ∂Ω,φ=φ0 on ∂Ω.$begin{cases}Aleft(uright)=rho left(uright)vert nabla varphi {vert }^{2},text{in},{Omega},quad hfill text{div}left(rho left(uright)nabla varphi right)=0,text{in},{Omega},quad hfill u=0,text{on},partial {Omega},quad hfill varphi ={varphi }_{0},text{on},partial {Omega}.quad hfill end{cases}$ where A(u) = −div a(x, u, ∇u) is a Leray-Lions operator of order p. The second member of the first equation is only in L1(Ω). We prove the existence of a bilateral solution by an approximation procedure, the keypoint being a penalization technique.
本文分析了以下非线性椭圆问题 A ( u ) = ρ ( u ) |∇ φ | 2 in Ω , div ( ρ ( u )∇ φ ) = 0 in Ω , u = 0 on ∂ Ω , φ = φ 0 on ∂ Ω 。 $begin{cases}Aleft(uright)=rho left(uright)vert nabla varphi {vert }^{2},text{in},{Omega},quad hfill text{div}left(rho left(uright)nabla varphi right)=0、u=0text{on}partial {Omega}quadhfill varphi ={varphi }_{0}text{on}partial {Omega}.其中 A(u) = -div a(x, u,∇u) 是阶数为 p 的勒雷-狮子算子。我们通过近似过程证明了双边解的存在，关键点在于惩罚技术。
{"title":"Increase of power leads to a bilateral solution to a strongly nonlinear elliptic coupled system","authors":"Francisco Ortegón Gallego, Mohamed Rhoudaf, Hajar Talbi","doi":"10.1515/ans-2023-0133","DOIUrl":"https://doi.org/10.1515/ans-2023-0133","url":null,"abstract":"In this paper, we analyze the following nonlinear elliptic problem <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\" overflow=\"scroll\"> <m:mfenced close=\"\" open=\"{\"> <m:mrow> <m:mtable> <m:mtr> <m:mtd columnalign=\"left\"> <m:mi>A</m:mi> <m:mrow> <m:mo stretchy=\"false\">(</m:mo> <m:mrow> <m:mi>u</m:mi> </m:mrow> <m:mo stretchy=\"false\">)</m:mo> </m:mrow> <m:mo>=</m:mo> <m:mi>ρ</m:mi> <m:mrow> <m:mo stretchy=\"false\">(</m:mo> <m:mrow> <m:mi>u</m:mi> </m:mrow> <m:mo stretchy=\"false\">)</m:mo> </m:mrow> <m:mo stretchy=\"false\">|</m:mo> <m:mi>∇</m:mi> <m:mi>φ</m:mi> <m:msup> <m:mrow> <m:mo stretchy=\"false\">|</m:mo> </m:mrow> <m:mrow> <m:mn>2</m:mn> </m:mrow> </m:msup> <m:mtext> in </m:mtext> <m:mi mathvariant=\"normal\">Ω</m:mi> <m:mo>,</m:mo> <m:mspace width=\"1em\" /> </m:mtd> </m:mtr> <m:mtr> <m:mtd columnalign=\"left\"> <m:mtext>div</m:mtext> <m:mrow> <m:mo stretchy=\"false\">(</m:mo> <m:mrow> <m:mi>ρ</m:mi> <m:mrow> <m:mo stretchy=\"false\">(</m:mo> <m:mrow> <m:mi>u</m:mi> </m:mrow> <m:mo stretchy=\"false\">)</m:mo> </m:mrow> <m:mi>∇</m:mi> <m:mi>φ</m:mi> </m:mrow> <m:mo stretchy=\"false\">)</m:mo> </m:mrow> <m:mo>=</m:mo> <m:mn>0</m:mn> <m:mtext> in </m:mtext> <m:mi mathvariant=\"normal\">Ω</m:mi> <m:mo>,</m:mo> <m:mspace width=\"1em\" /> </m:mtd> </m:mtr> <m:mtr> <m:mtd columnalign=\"left\"> <m:mi>u</m:mi> <m:mo>=</m:mo> <m:mn>0</m:mn> <m:mtext> on </m:mtext> <m:mi>∂</m:mi> <m:mi mathvariant=\"normal\">Ω</m:mi> <m:mo>,</m:mo> <m:mspace width=\"1em\" /> </m:mtd> </m:mtr> <m:mtr> <m:mtd columnalign=\"left\"> <m:mi>φ</m:mi> <m:mo>=</m:mo> <m:msub> <m:mrow> <m:mi>φ</m:mi> </m:mrow> <m:mrow> <m:mn>0</m:mn> </m:mrow> </m:msub> <m:mtext> on </m:mtext> <m:mi>∂</m:mi> <m:mi mathvariant=\"normal\">Ω</m:mi> <m:mo>.</m:mo> <m:mspace width=\"1em\" /> </m:mtd> </m:mtr> </m:mtable> </m:mrow> </m:mfenced> </m:math> <jats:tex-math>$begin{cases}Aleft(uright)=rho left(uright)vert nabla varphi {vert }^{2},text{in},{Omega},quad hfill text{div}left(rho left(uright)nabla varphi right)=0,text{in},{Omega},quad hfill u=0,text{on},partial {Omega},quad hfill varphi ={varphi }_{0},text{on},partial {Omega}.quad hfill end{cases}$</jats:tex-math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_ans-2023-0133_ineq_001.png\" /> </jats:alternatives> </jats:inline-formula> where <jats:italic>A</jats:italic>(<jats:italic>u</jats:italic>) = −div <jats:italic>a</jats:italic>(<jats:italic>x</jats:italic>, <jats:italic>u</jats:italic>, ∇<jats:italic>u</jats:italic>) is a Leray-Lions operator of order <jats:italic>p</jats:italic>. The second member of the first equation is only in <jats:italic>L</jats:italic> <jats:sup>1</jats:sup>(Ω). We prove the existence of a bilateral solution by an approximation procedure, the keypoint being a penalization technique.","PeriodicalId":7191,"journal":{"name":"Advanced Nonlinear Studies","volume":"9 1","pages":""},"PeriodicalIF":1.8,"publicationDate":"2024-04-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140624049","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this paper we prove gradient estimates of both elliptic and parabolic types, specifically, of Souplet-Zhang, Hamilton and Li-Yau types for positive smooth solutions to a class of nonlinear parabolic equations involving the Witten or drifting Laplacian on smooth metric measure spaces. These estimates are established under various curvature conditions and lower bounds on the generalised Bakry-Émery Ricci tensor and find utility in proving elliptic and parabolic Harnack-type inequalities as well as general Liouville-type and other global constancy results. Several applications and consequences are presented and discussed.
{"title":"Curvature conditions, Liouville-type theorems and Harnack inequalities for a nonlinear parabolic equation on smooth metric measure spaces","authors":"Ali Taheri, Vahideh Vahidifar","doi":"10.1515/ans-2023-0120","DOIUrl":"https://doi.org/10.1515/ans-2023-0120","url":null,"abstract":"In this paper we prove gradient estimates of both elliptic and parabolic types, specifically, of Souplet-Zhang, Hamilton and Li-Yau types for positive smooth solutions to a class of nonlinear parabolic equations involving the Witten or drifting Laplacian on smooth metric measure spaces. These estimates are established under various curvature conditions and lower bounds on the generalised Bakry-Émery Ricci tensor and find utility in proving elliptic and parabolic Harnack-type inequalities as well as general Liouville-type and other global constancy results. Several applications and consequences are presented and discussed.","PeriodicalId":7191,"journal":{"name":"Advanced Nonlinear Studies","volume":"2 1","pages":""},"PeriodicalIF":1.8,"publicationDate":"2024-04-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140579549","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
我们重点研究以下分式 (p, q) - 肖卡问题： ( - Δ ) p s u + ( - Δ ) q s u + V ( ε x ) ( | u | p - 2 u + | u | q - 2 u ) = 1 | x | μ * F ( u ) f ( u ) in R N , u ∈ W s , p ( R N ) ∩ W s , q ( R N ) , u >；在 R N 中为 0 、 $begin{cases}{left(-{Delta}right)}_{p}^{s}u+{left(-{Delta}right)}_{q}^{s}u+Vleft(varepsilon xright)left(vert u{vert }^{p-2}u+vert u{vert }^{q-2}uright)=left(frac{1}{vert x{vert }^{mu }}{ast}Fleft(uright)right)fleft(uright)、uin {W}^{s,p}left({mathbb{R}}^{N}right)cap {W}^{s,q}left({mathbb{R}}^{N}right), u{ >；}0text{in},{mathbb{R}}^{N},quadhfillend{cases}$ 其中 ɛ > 0 是一个小参数，0 < s < 1, 1 < p < q < N s $1{< }p{< }q{<；}frac{N}{s}$ , 0 < μ < sp, ( - Δ ) r s ${left(-{Delta}right)}_{r}^{s}$ , r∈ {p, q}, 是分数 r 拉普拉斯算子, V ： R N → R $V:{mathbb{R}}^{N}to mathbb{R}$ 是满足局部条件的正连续势，f ：R → R $f:mathbb{R}tomathbb{R}$ 是一个连续的非线性，在无穷远处有亚临界增长，并且 F ( t ) = ∫ 0 t f ( τ ) d τ $Fleft(tright)={int }_{0}^{t}fleft(tau right) mathrm{d}tau $ 。应用合适的变分法和拓扑法，我们将解的数量与势 V 达到最小值的集合的拓扑关系联系起来。
In this paper, we consider the <jats:italic>k</jats:italic>-Hessian problem <jats:italic>S</jats:italic> <jats:sub> <jats:italic>k</jats:italic> </jats:sub>(<jats:italic>D</jats:italic> <jats:sup>2</jats:sup> <jats:italic>u</jats:italic>) = <jats:italic>b</jats:italic>(<jats:italic>x</jats:italic>)<jats:italic>f</jats:italic>(<jats:italic>u</jats:italic>) in Ω, <jats:italic>u</jats:italic> = +∞ on <jats:italic>∂</jats:italic>Ω, where Ω is a <jats:italic>C</jats:italic> <jats:sup>∞</jats:sup>-smooth bounded strictly (<jats:italic>k</jats:italic> − 1)-convex domain in <jats:inline-formula> <jats:alternatives> <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <m:msup> <m:mrow> <m:mi mathvariant="double-struck">R</m:mi> </m:mrow> <m:mrow> <m:mi>N</m:mi> </m:mrow> </m:msup> </m:math> <jats:tex-math>${mathbb{R}}^{N}$</jats:tex-math> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_ans-2023-0128_ineq_001.png" /> </jats:alternatives> </jats:inline-formula> with <jats:italic>N</jats:italic> ≥ 2, <jats:italic>b</jats:italic> ∈ C<jats:sup>∞</jats:sup>(Ω) is positive in Ω and may be singular or vanish on <jats:italic>∂</jats:italic>Ω, <jats:italic>f</jats:italic> ∈ <jats:italic>C</jats:italic>[0, ∞) ∩ <jats:italic>C</jats:italic> <jats:sup>1</jats:sup>(0, ∞) (or <jats:inline-formula> <jats:alternatives> <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <m:mi>f</m:mi> <m:mo>∈</m:mo> <m:msup> <m:mrow> <m:mi>C</m:mi> </m:mrow> <m:mrow> <m:mn>1</m:mn> </m:mrow> </m:msup> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mrow> <m:mi mathvariant="double-struck">R</m:mi> </m:mrow> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:math> <jats:tex-math>$fin {C}^{1}left(mathbb{R}right)$</jats:tex-math> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_ans-2023-0128_ineq_002.png" /> </jats:alternatives> </jats:inline-formula>) is a positive and increasing function. We establish the first expansions (equalities) of <jats:italic>k</jats:italic>-convex solutions to the above problem when <jats:italic>f</jats:italic> is borderline regularly varying and Γ-varying at infinity respectively. For the former, we reveal the exact influences of some indexes of <jats:italic>f</jats:italic> and principal curvatures of <jats:italic>∂</jats:italic>Ω on the first expansion of solutions. For the latter, we find the principal curvatures of <jats:italic>∂</jats:italic>Ω have no influences on the expansions. Our results and methods are quite different from the existing ones (including <jats:italic>k</jats:italic> = <jats:italic>N</jats:italic>). Moreover, we know the existence of <jats:italic>k</jats:italic>-convex solutions to the above problem (including <jats:italic>k</jats:italic> = <jats:italic>N</jats:italic>) is still an open problem when <jats:italic>b</jats:italic> possesses high singularity on <jats:italic>∂</jats:italic>Ω and <jats:italic>f</jats:italic> satisfies Keller–Os
本文考虑 k-Hessian 问题 S k (D 2 u) = b(x)f(u) in Ω, u = +∞ on ∂Ω，其中 Ω 是 R N ${mathbb{R}}^{N}$ 中的一个 C ∞-光滑有界严格（k - 1）-凸域，N ≥ 2、b∈ C∞(Ω) 在 Ω 中为正，在 ∂Ω 上可能是奇异的或消失，f ∈ C[0, ∞) ∩ C 1(0, ∞) （或 f∈ C 1 ( R ) $fin {C}^{1}left(mathbb{R}right)$ ）是一个正的递增函数。我们分别建立了当 f 在无穷远处为边界正则变化和 Γ 变化时，上述问题的 k 个凸解的第一次展开（相等）。对于前者，我们揭示了 f 的某些指数和 ∂Ω 的主曲率对解的第一次展开的确切影响。对于后者，我们发现∂Ω 的主曲率对展开没有影响。我们的结果和方法与现有的结果和方法（包括 k = N）截然不同。此外，我们还知道，当 b 在 ∂Ω 上具有高奇异性且 f 满足 Keller-Osserman 类型条件时，上述问题（包括 k = N）的 k 凸解的存在仍是一个未决问题。对于球中的径向对称情况，我们给出了这个开放问题的肯定答案，然后进一步展示了所有径向大解的全局估计。
{"title":"On the large solutions to a class of k-Hessian problems","authors":"Haitao Wan","doi":"10.1515/ans-2023-0128","DOIUrl":"https://doi.org/10.1515/ans-2023-0128","url":null,"abstract":"In this paper, we consider the <jats:italic>k</jats:italic>-Hessian problem <jats:italic>S</jats:italic> <jats:sub> <jats:italic>k</jats:italic> </jats:sub>(<jats:italic>D</jats:italic> <jats:sup>2</jats:sup> <jats:italic>u</jats:italic>) = <jats:italic>b</jats:italic>(<jats:italic>x</jats:italic>)<jats:italic>f</jats:italic>(<jats:italic>u</jats:italic>) in Ω, <jats:italic>u</jats:italic> = +∞ on <jats:italic>∂</jats:italic>Ω, where Ω is a <jats:italic>C</jats:italic> <jats:sup>∞</jats:sup>-smooth bounded strictly (<jats:italic>k</jats:italic> − 1)-convex domain in <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\" overflow=\"scroll\"> <m:msup> <m:mrow> <m:mi mathvariant=\"double-struck\">R</m:mi> </m:mrow> <m:mrow> <m:mi>N</m:mi> </m:mrow> </m:msup> </m:math> <jats:tex-math>${mathbb{R}}^{N}$</jats:tex-math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_ans-2023-0128_ineq_001.png\" /> </jats:alternatives> </jats:inline-formula> with <jats:italic>N</jats:italic> ≥ 2, <jats:italic>b</jats:italic> ∈ C<jats:sup>∞</jats:sup>(Ω) is positive in Ω and may be singular or vanish on <jats:italic>∂</jats:italic>Ω, <jats:italic>f</jats:italic> ∈ <jats:italic>C</jats:italic>[0, ∞) ∩ <jats:italic>C</jats:italic> <jats:sup>1</jats:sup>(0, ∞) (or <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\" overflow=\"scroll\"> <m:mi>f</m:mi> <m:mo>∈</m:mo> <m:msup> <m:mrow> <m:mi>C</m:mi> </m:mrow> <m:mrow> <m:mn>1</m:mn> </m:mrow> </m:msup> <m:mrow> <m:mo stretchy=\"false\">(</m:mo> <m:mrow> <m:mi mathvariant=\"double-struck\">R</m:mi> </m:mrow> <m:mo stretchy=\"false\">)</m:mo> </m:mrow> </m:math> <jats:tex-math>$fin {C}^{1}left(mathbb{R}right)$</jats:tex-math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_ans-2023-0128_ineq_002.png\" /> </jats:alternatives> </jats:inline-formula>) is a positive and increasing function. We establish the first expansions (equalities) of <jats:italic>k</jats:italic>-convex solutions to the above problem when <jats:italic>f</jats:italic> is borderline regularly varying and Γ-varying at infinity respectively. For the former, we reveal the exact influences of some indexes of <jats:italic>f</jats:italic> and principal curvatures of <jats:italic>∂</jats:italic>Ω on the first expansion of solutions. For the latter, we find the principal curvatures of <jats:italic>∂</jats:italic>Ω have no influences on the expansions. Our results and methods are quite different from the existing ones (including <jats:italic>k</jats:italic> = <jats:italic>N</jats:italic>). Moreover, we know the existence of <jats:italic>k</jats:italic>-convex solutions to the above problem (including <jats:italic>k</jats:italic> = <jats:italic>N</jats:italic>) is still an open problem when <jats:italic>b</jats:italic> possesses high singularity on <jats:italic>∂</jats:italic>Ω and <jats:italic>f</jats:italic> satisfies Keller–Os","PeriodicalId":7191,"journal":{"name":"Advanced Nonlinear Studies","volume":"49 1","pages":""},"PeriodicalIF":1.8,"publicationDate":"2024-04-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140579435","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
在本文中，我们首先基于 Brendle、Guan 和 Li 引入的局部约束反曲率流（"An inverse curvature type hypersurface flow in H n + 1 ${{mathbb{H}}^{n+1}$ ," (Preprint) ），建立并验证了双曲空间 H n + 1 ${{mathbb{H}}^{n+1}$ 中平均曲率的新的尖锐双曲版 Michael-Simon 不等式，如下 (0.1) ∫ M λ ′ f 2 E 1 2 + | ∇ M f | 2 - ∫ M∇ ̄ f λ ′ 、ν + ∫ ∂ M f ≥ ω n 1 n ∫ M f n n - 1 n - 1 n $ $underset{M}{int }{lambda }^{prime }sqrt{{f}^{2}{E}_{1}^{2}+vert {nabla }^{M}f{vert }^{2}}-ungeerset{M}{int }langle {barnabla }left(f{lambda }^{prime }right)、nu rangle +underset{partial M}{int }fge {omega }_{n}^{frac{1}{n}}{left(underset{M}{int }{f}^{frac{n}{n-1}}right)}^{frac{n-1}{n}}$$ 前提是 M 是 h-convex 且 f 是正的平滑函数、其中 λ′(r) = coshr。特别是，当 f 为常数时，(0.1) 与 Brendle、Hung 和 Wang 在（"A Minkowski inequality for hypersurfaces in the anti-de Sitter-Schwarzschild manifold," Commun.纯应用数学》，第 69 卷，第 1 期，第 124-144 页，2016 年）。此外，我们还通过布伦德尔-关-李流（"An inverse curvature type hypersurface flow in H n + 1 ${mathbb{H}}^{n+1}$ ," （Preprint））建立并证实了H n + 1 ${mathbb{H}}^{n+1}$ 中第k次均值曲率的新的尖锐迈克尔-西蒙不等式（0.2) ∫ M λ ′ f 2 E k 2 + |∇ M f | 2 E k - 1 2 - ∫ M∇ ̄ f λ ′ , ν ⋅ E k - 1 + ∫∂ M f ⋅ E k - 1 ≥ p k ◦ q 1 - 1 ( W 1 ( Ω ) ) 1 n - k + 1 ∫ M f n - k + 1 n - k ⋅ E k - 1 n - k n - k + 1 begin{align}hfill &；underset{M}{int }{lambda }^{prime } （sqrt{{f}^{2}{E}_{k}^{2}+vert {nabla }^{M}f{vert }^{2}{E}_{k-1}^{2}}-underset{M}{int }langle バッグ {nabla } （left(f{lambda }^{prime }right)、nu rangle cdot {E}_{k-1}+underset{partial M}{int }fcdot {E}_{k-1}hfill hfill &；quad ge {left({p}_{k}{circ}{q}_{1}^{-1}left({W}_{1}left({Omega}right)right)right)}^{frac{1}{n-k+1}}{left(underset{M}{int }{f}^{frac{n-k+1}{n-k}}{cdot{E}_{k-1}/right)}^{frac{n-k}{n-k+1}}hfill （end{align}），前提是 M 是 h-vex 的，并且 Ω 是 M 所包围的域、p k (r) = ω n (λ′) k-1、 W 1 ( ω ) = 1 n | M | ${W}_{1}left({Omega}right)=frac{1}{n}vert Mvert $ , λ′(r) = coshr, q 1 ( r ) = W 1 S r n + 1 ${q}_{1}left(rright)={W}_{1}left({S}_{r}^{n+1}right)$ 、是半径为 r 的大地球体的面积，而 q 1 - 1 ${q}_{1}^{-1}$ 是 q 1 的反函数。特别是，当 f 为常数且 k 为奇数时，(0.2) 正是胡、李和魏在《双曲空间中的局部约束曲率流和几何不等式》（"Locally constrained curvature flows and geometric inequalities in hyperbolic space"）一文中证明的加权亚历山德罗夫-芬切尔不等式。3-4, pp.）
{"title":"Michael-Simon type inequalities in hyperbolic space H n + 1 ${mathbb{H}}^{n+1}$ via Brendle-Guan-Li’s flows","authors":"Jingshi Cui, Peibiao Zhao","doi":"10.1515/ans-2023-0127","DOIUrl":"https://doi.org/10.1515/ans-2023-0127","url":null,"abstract":"In the present paper, we first establish and verify a new sharp hyperbolic version of the Michael-Simon inequality for mean curvatures in hyperbolic space <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\" overflow=\"scroll\"> <m:msup> <m:mrow> <m:mi mathvariant=\"double-struck\">H</m:mi> </m:mrow> <m:mrow> <m:mi>n</m:mi> <m:mo>+</m:mo> <m:mn>1</m:mn> </m:mrow> </m:msup> </m:math> <jats:tex-math>${mathbb{H}}^{n+1}$</jats:tex-math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_ans-2023-0127_ineq_002.png\" /> </jats:alternatives> </jats:inline-formula> based on the locally constrained inverse curvature flow introduced by Brendle, Guan and Li (“An inverse curvature type hypersurface flow in <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\" overflow=\"scroll\"> <m:msup> <m:mrow> <m:mi mathvariant=\"double-struck\">H</m:mi> </m:mrow> <m:mrow> <m:mi>n</m:mi> <m:mo>+</m:mo> <m:mn>1</m:mn> </m:mrow> </m:msup> </m:math> <jats:tex-math>${mathbb{H}}^{n+1}$</jats:tex-math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_ans-2023-0127_ineq_003.png\" /> </jats:alternatives> </jats:inline-formula>,” (Preprint)) as follows<jats:disp-formula> <jats:label>(0.1)</jats:label> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\" display=\"block\" overflow=\"scroll\"> <m:munder> <m:mrow> <m:mo>∫</m:mo> </m:mrow> <m:mrow> <m:mi>M</m:mi> </m:mrow> </m:munder> <m:msup> <m:mrow> <m:mi>λ</m:mi> </m:mrow> <m:mrow> <m:mo>′</m:mo> </m:mrow> </m:msup> <m:msqrt> <m:mrow> <m:msup> <m:mrow> <m:mi>f</m:mi> </m:mrow> <m:mrow> <m:mn>2</m:mn> </m:mrow> </m:msup> <m:msubsup> <m:mrow> <m:mi>E</m:mi> </m:mrow> <m:mrow> <m:mn>1</m:mn> </m:mrow> <m:mrow> <m:mn>2</m:mn> </m:mrow> </m:msubsup> <m:mo>+</m:mo> <m:mo stretchy=\"false\">|</m:mo> <m:msup> <m:mrow> <m:mi>∇</m:mi> </m:mrow> <m:mrow> <m:mi>M</m:mi> </m:mrow> </m:msup> <m:mi>f</m:mi> <m:msup> <m:mrow> <m:mo stretchy=\"false\">|</m:mo> </m:mrow> <m:mrow> <m:mn>2</m:mn> </m:mrow> </m:msup> </m:mrow> </m:msqrt> <m:mo>−</m:mo> <m:munder> <m:mrow> <m:mo>∫</m:mo> </m:mrow> <m:mrow> <m:mi>M</m:mi> </m:mrow> </m:munder> <m:mfenced close=\"⟩\" open=\"⟨\"> <m:mrow> <m:mrow> <m:mover accent=\"true\"> <m:mrow> <m:mi>∇</m:mi> </m:mrow> <m:mo>̄</m:mo> </m:mover> </m:mrow> <m:mfenced close=\")\" open=\"(\"> <m:mrow> <m:mi>f</m:mi> <m:msup> <m:mrow> <m:mi>λ</m:mi> </m:mrow> <m:mrow> <m:mo>′</m:mo> </m:mrow> </m:msup> </m:mrow> </m:mfenced> <m:mo>,</m:mo> <m:mi>ν</m:mi> </m:mrow> </m:mfenced> <m:mo>+</m:mo> <m:munder> <m:mrow> <m:mo>∫</m:mo> </m:mrow> <m:mrow> <m:mi>∂</m:mi> <m:mi>M</m:mi> </m:mrow> </m:munder> <m:mi>f</m:mi> <m:mo>≥</m:mo> <m:msubsup> <m:mrow> <m:mi>ω</m:mi> </m:mrow> <m:mrow> <m:mi>n</m:mi> </m:mrow> <m:mrow> <m:mfrac> <m:mrow> <m:mn>1</m:mn> </m:mrow> <m:mrow> <m:mi>n</m:mi> </m:mrow> </m:mfrac> </m:mrow> </m:msubsup> <m:msup> <m:mrow> <m:mfenced close=\")\" open=\"(\"> <m:mrow","PeriodicalId":7191,"journal":{"name":"Advanced Nonlinear Studies","volume":"32 1","pages":""},"PeriodicalIF":1.8,"publicationDate":"2024-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140579431","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this paper, we summarize some of the recent developments in the area of fractional elliptic and parabolic equations with focus on how to apply the sliding method and the method of moving planes to obtain qualitative properties of solutions. We will compare the two methods and point out the pros and cons of each. We will demonstrate how to modify the ideas and techniques in studying fractional elliptic equations and then to employ them to investigate fractional parabolic problems. Besides deriving monotonicity of solutions, some other applications of the sliding method will be illustrated. These results have more or less appeared in a series of previous literatures, in which the ideas were usually submerged in detailed calculations. What we are trying to do here is to single out these ideas and illuminate the inner connections among them by using figures and intuitive languages, so that the readers can see the whole picture and quickly grasp the essence of these useful methods and will be able to apply them to solve a variety of other fractional elliptic and parabolic problems.
{"title":"Moving planes and sliding methods for fractional elliptic and parabolic equations","authors":"Wenxiong Chen, Yeyao Hu, Lingwei Ma","doi":"10.1515/ans-2022-0069","DOIUrl":"https://doi.org/10.1515/ans-2022-0069","url":null,"abstract":"In this paper, we summarize some of the recent developments in the area of fractional elliptic and parabolic equations with focus on how to apply the sliding method and the method of moving planes to obtain qualitative properties of solutions. We will compare the two methods and point out the pros and cons of each. We will demonstrate how to modify the ideas and techniques in studying fractional elliptic equations and then to employ them to investigate fractional parabolic problems. Besides deriving monotonicity of solutions, some other applications of the sliding method will be illustrated. These results have more or less appeared in a series of previous literatures, in which the ideas were usually submerged in detailed calculations. What we are trying to do here is to single out these ideas and illuminate the inner connections among them by using figures and intuitive languages, so that the readers can see the whole picture and quickly grasp the essence of these useful methods and will be able to apply them to solve a variety of other fractional elliptic and parabolic problems.","PeriodicalId":7191,"journal":{"name":"Advanced Nonlinear Studies","volume":"1 1","pages":""},"PeriodicalIF":1.8,"publicationDate":"2024-03-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140168008","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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