Pub Date : 2023-11-03DOI: 10.1007/s10013-023-00653-z
Martin Dindoš, Erik Sätterqvist, Martin Ulmer
Abstract In the present paper we study perturbation theory for the $$L^p$$ Lp Dirichlet problem on bounded chord arc domains for elliptic operators in divergence form with potentially unbounded antisymmetric part in BMO. Specifically, given elliptic operators $$L_0 = text {div}(A_0nabla )$$ L0=div(A0∇) and $$L_1 = text {div}(A_1nabla )$$ L1=div(A1∇) such that the $$L^p$$ Lp Dirichlet problem for $$L_0$$ L0 is solvable for some $$p>1$$ p>1 ; we show that if $$A_0 - A_1$$ A0-A1 satisfies certain Carleson condition, then the $$L^q$$ Lq Dirichlet problem for $$L_1$$ L1 is solvable for some $$q ge p$$ q≥p . Moreover if the Carleson norm is small then we may take $$q=p$$ q=p . We use the approach first introduced in Fefferman–Kenig–Pipher ’91 on the unit ball, and build on Milakis–Pipher–Toro ’11 where the large norm case was shown for symmetric matrices on bounded chord arc domains. We then apply this to solve the $$L^p$$ Lp
摘要本文研究了BMO中具有潜在无界反对称部分的发散型椭圆算子在有界弦弧域上的$$L^p$$ L p Dirichlet问题的摄动理论。具体地说，给定椭圆算子$$L_0 = text {div}(A_0nabla )$$ L 0 = div (A 0∇)和$$L_1 = text {div}(A_1nabla )$$ L 1 = div (A 1∇)使得$$L_0$$ L 0的$$L^p$$ L p Dirichlet问题对于某些$$p>1$$ p ＆gt是可解的;1;我们证明了如果$$A_0 - A_1$$ A 0 - a1满足一定的Carleson条件，那么$$L_1$$ L 1的$$L^q$$ L q Dirichlet问题对于某些$$q ge p$$ q≥p是可解的。此外，如果Carleson范数很小，那么我们可以取$$q=p$$ q = p。我们使用fefferman - keng - piphher ' 91在单位球上首次引入的方法，并建立在Milakis-Pipher-Toro ' 11的基础上，其中展示了有界弦弧域上对称矩阵的大范数情况。然后，我们将此应用于求解算子$$L = text {div}(Anabla )$$ L = div (a∇)在有界Lipschitz域上的$$L^p$$ L p Dirichlet问题，其中a满足类似于kenig - piphher ' 01和Dindoš-Petermichl-Pipher ' 07中假设的Carleson条件，但具有无界反对称部分。
{"title":"Perturbation Theory for Second Order Elliptic Operators with BMO Antisymmetric Part","authors":"Martin Dindoš, Erik Sätterqvist, Martin Ulmer","doi":"10.1007/s10013-023-00653-z","DOIUrl":"https://doi.org/10.1007/s10013-023-00653-z","url":null,"abstract":"Abstract In the present paper we study perturbation theory for the $$L^p$$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:msup> <mml:mi>L</mml:mi> <mml:mi>p</mml:mi> </mml:msup> </mml:math> Dirichlet problem on bounded chord arc domains for elliptic operators in divergence form with potentially unbounded antisymmetric part in BMO. Specifically, given elliptic operators $$L_0 = text {div}(A_0nabla )$$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mrow> <mml:msub> <mml:mi>L</mml:mi> <mml:mn>0</mml:mn> </mml:msub> <mml:mo>=</mml:mo> <mml:mtext>div</mml:mtext> <mml:mrow> <mml:mo>(</mml:mo> <mml:msub> <mml:mi>A</mml:mi> <mml:mn>0</mml:mn> </mml:msub> <mml:mi>∇</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:mrow> </mml:math> and $$L_1 = text {div}(A_1nabla )$$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mrow> <mml:msub> <mml:mi>L</mml:mi> <mml:mn>1</mml:mn> </mml:msub> <mml:mo>=</mml:mo> <mml:mtext>div</mml:mtext> <mml:mrow> <mml:mo>(</mml:mo> <mml:msub> <mml:mi>A</mml:mi> <mml:mn>1</mml:mn> </mml:msub> <mml:mi>∇</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:mrow> </mml:math> such that the $$L^p$$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:msup> <mml:mi>L</mml:mi> <mml:mi>p</mml:mi> </mml:msup> </mml:math> Dirichlet problem for $$L_0$$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:msub> <mml:mi>L</mml:mi> <mml:mn>0</mml:mn> </mml:msub> </mml:math> is solvable for some $$p>1$$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mrow> <mml:mi>p</mml:mi> <mml:mo>></mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:math> ; we show that if $$A_0 - A_1$$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mrow> <mml:msub> <mml:mi>A</mml:mi> <mml:mn>0</mml:mn> </mml:msub> <mml:mo>-</mml:mo> <mml:msub> <mml:mi>A</mml:mi> <mml:mn>1</mml:mn> </mml:msub> </mml:mrow> </mml:math> satisfies certain Carleson condition, then the $$L^q$$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:msup> <mml:mi>L</mml:mi> <mml:mi>q</mml:mi> </mml:msup> </mml:math> Dirichlet problem for $$L_1$$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:msub> <mml:mi>L</mml:mi> <mml:mn>1</mml:mn> </mml:msub> </mml:math> is solvable for some $$q ge p$$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mrow> <mml:mi>q</mml:mi> <mml:mo>≥</mml:mo> <mml:mi>p</mml:mi> </mml:mrow> </mml:math> . Moreover if the Carleson norm is small then we may take $$q=p$$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mrow> <mml:mi>q</mml:mi> <mml:mo>=</mml:mo> <mml:mi>p</mml:mi> </mml:mrow> </mml:math> . We use the approach first introduced in Fefferman–Kenig–Pipher ’91 on the unit ball, and build on Milakis–Pipher–Toro ’11 where the large norm case was shown for symmetric matrices on bounded chord arc domains. We then apply this to solve the $$L^p$$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:msup> <mml:mi>L</mml:mi> <mml:mi>p</","PeriodicalId":45919,"journal":{"name":"Vietnam Journal of Mathematics","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-11-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135818801","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-10-10DOI: 10.1007/s10013-023-00656-w
Alexandru D. Ionescu, Hao Jia
{"title":"On the Stability of Shear Flows in Bounded Channels, I: Monotonic Shear Flows","authors":"Alexandru D. Ionescu, Hao Jia","doi":"10.1007/s10013-023-00656-w","DOIUrl":"https://doi.org/10.1007/s10013-023-00656-w","url":null,"abstract":"","PeriodicalId":45919,"journal":{"name":"Vietnam Journal of Mathematics","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-10-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136294979","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-10-07DOI: 10.1007/s10013-023-00655-x
Nguyen Khoa Son, Nguyen Thi Hong
{"title":"Absolute Exponential Stability Criteria for Some Classes of Nonlinear Time-Varying Systems with Delays and Sector Nonlinearities","authors":"Nguyen Khoa Son, Nguyen Thi Hong","doi":"10.1007/s10013-023-00655-x","DOIUrl":"https://doi.org/10.1007/s10013-023-00655-x","url":null,"abstract":"","PeriodicalId":45919,"journal":{"name":"Vietnam Journal of Mathematics","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-10-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135253342","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-10-04DOI: 10.1007/s10013-023-00652-0
Boulos El Hilany, Elias Tsigaridas
Abstract We introduce novel mathematical and computational tools to develop a complete algorithm for computing the set of non-properness of polynomials maps in the plane. In particular, this set, which we call the Jelonek set , is a subset of $$mathbb {K}^2$$ K2 , where a dominant polynomial map $$f: mathbb {K}^2 rightarrow mathbb {K}^2$$ f:K2→K2 is not proper; $$mathbb {K}$$ K could be either $$mathbb {C}$$ C or $$mathbb {R}$$ R . Unlike all the previously known approaches we make no assumptions on f whenever $$mathbb {K} = mathbb {R}$$ K=R ; this is the first algorithm with this property. The algorithm takes into account the Newton polytopes of the polynomials. As a byproduct we provide a finer representation of the set of non-properness as a union of semi-algebraic curves, that correspond to edges of the Newton polytopes, which is of independent interest. Finally, we present a precise Boolean complexity analysis of the algorithm and a prototype implementation in maple .
摘要引入新的数学和计算工具，提出了一种计算平面上多项式映射非适当集的完整算法。特别地，这个集合，我们称之为Jelonek集合，是$$mathbb {K}^2$$ K 2的一个子集，其中一个优势多项式映射$$f: mathbb {K}^2 rightarrow mathbb {K}^2$$ f: K 2→K 2是不合适的;$$mathbb {K}$$ K可以是$$mathbb {C}$$ C或者$$mathbb {R}$$ R。不像以前所有已知的方法，当$$mathbb {K} = mathbb {R}$$ K = R时，我们不对f做任何假设;这是第一个具有这种性质的算法。该算法考虑了多项式的牛顿多面体。作为一个副产品，我们提供了一个更精细的非适当性集合的表示作为半代数曲线的并，这些曲线对应于牛顿多面体的边，这是一个独立的兴趣。最后，我们对该算法进行了精确的布尔复杂度分析，并在maple中实现了原型。
{"title":"Computing the Non-properness Set of Real Polynomial Maps in the Plane","authors":"Boulos El Hilany, Elias Tsigaridas","doi":"10.1007/s10013-023-00652-0","DOIUrl":"https://doi.org/10.1007/s10013-023-00652-0","url":null,"abstract":"Abstract We introduce novel mathematical and computational tools to develop a complete algorithm for computing the set of non-properness of polynomials maps in the plane. In particular, this set, which we call the Jelonek set , is a subset of $$mathbb {K}^2$$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:msup> <mml:mrow> <mml:mi>K</mml:mi> </mml:mrow> <mml:mn>2</mml:mn> </mml:msup> </mml:math> , where a dominant polynomial map $$f: mathbb {K}^2 rightarrow mathbb {K}^2$$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mrow> <mml:mi>f</mml:mi> <mml:mo>:</mml:mo> <mml:msup> <mml:mrow> <mml:mi>K</mml:mi> </mml:mrow> <mml:mn>2</mml:mn> </mml:msup> <mml:mo>→</mml:mo> <mml:msup> <mml:mrow> <mml:mi>K</mml:mi> </mml:mrow> <mml:mn>2</mml:mn> </mml:msup> </mml:mrow> </mml:math> is not proper; $$mathbb {K}$$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mi>K</mml:mi> </mml:math> could be either $$mathbb {C}$$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mi>C</mml:mi> </mml:math> or $$mathbb {R}$$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mi>R</mml:mi> </mml:math> . Unlike all the previously known approaches we make no assumptions on f whenever $$mathbb {K} = mathbb {R}$$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mrow> <mml:mi>K</mml:mi> <mml:mo>=</mml:mo> <mml:mi>R</mml:mi> </mml:mrow> </mml:math> ; this is the first algorithm with this property. The algorithm takes into account the Newton polytopes of the polynomials. As a byproduct we provide a finer representation of the set of non-properness as a union of semi-algebraic curves, that correspond to edges of the Newton polytopes, which is of independent interest. Finally, we present a precise Boolean complexity analysis of the algorithm and a prototype implementation in maple .","PeriodicalId":45919,"journal":{"name":"Vietnam Journal of Mathematics","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-10-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135592116","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-09-29DOI: 10.1007/s10013-023-00654-y
Blair Davey
{"title":"On Landis’ Conjecture in the Plane for Potentials with Growth","authors":"Blair Davey","doi":"10.1007/s10013-023-00654-y","DOIUrl":"https://doi.org/10.1007/s10013-023-00654-y","url":null,"abstract":"","PeriodicalId":45919,"journal":{"name":"Vietnam Journal of Mathematics","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-09-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135193140","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-09-26DOI: 10.1007/s10013-023-00648-w
Andrew Lawrie
Abstract In this expository note, we prove a localized bubbling result for solutions of the energy critical nonlinear heat equation with bounded $$dot{H} ^1$$ H˙1 norm. The proof uses a combination of Gérard’s profile decomposition (ESAIM Control Optim. Calc. Var. 3 : 213–233, 1998), concentration compactness techniques in the spirit of Duyckaerts, Kenig, and Merle’s seminal work (Geom. Funct. Anal. 22 : 639–698, 2012), and a virial argument in the spirit of Jia and Kenig’s work (Amer. J. Math. 139 : 1521–1603, 2017) to deduce the vanishing of the error in the neck regions between the bubbles. The argument is based closely on an analogous lemma proved in the author’s recent work with Jendrej (arXiv:2210.14963, 2022) on the equivariant harmonic map heat flow in dimension two.
摘要本文证明了具有有界$$dot{H} ^1$$ H˙1范数的能量临界非线性热方程解的一个局域冒泡结果。该证明使用了gsamim的配置文件分解(ESAIM Control Optim)的组合。Calc. Var. 3: 213 - 233,1998)，在Duyckaerts, Kenig和Merle的开创性工作的精神集中密实技术(Geom。函数。《论文集》，2012年第22期:639-698页)，以及在贾和柯尼格的作品精神中进行的一场病毒式辩论(美国)。[j] .数学学报，39(1):1521-1603,2017)，以推断气泡之间的颈部区域误差的消失。该论证紧密地基于作者最近与Jendrej (arXiv:2210.14963, 2022)在二维等变调和映射热流上证明的一个类似引理。
{"title":"Localized Sequential Bubbling for the Radial Energy Critical Semilinear Heat Equation","authors":"Andrew Lawrie","doi":"10.1007/s10013-023-00648-w","DOIUrl":"https://doi.org/10.1007/s10013-023-00648-w","url":null,"abstract":"Abstract In this expository note, we prove a localized bubbling result for solutions of the energy critical nonlinear heat equation with bounded $$dot{H} ^1$$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:msup> <mml:mover> <mml:mi>H</mml:mi> <mml:mo>˙</mml:mo> </mml:mover> <mml:mn>1</mml:mn> </mml:msup> </mml:math> norm. The proof uses a combination of Gérard’s profile decomposition (ESAIM Control Optim. Calc. Var. 3 : 213–233, 1998), concentration compactness techniques in the spirit of Duyckaerts, Kenig, and Merle’s seminal work (Geom. Funct. Anal. 22 : 639–698, 2012), and a virial argument in the spirit of Jia and Kenig’s work (Amer. J. Math. 139 : 1521–1603, 2017) to deduce the vanishing of the error in the neck regions between the bubbles. The argument is based closely on an analogous lemma proved in the author’s recent work with Jendrej (arXiv:2210.14963, 2022) on the equivariant harmonic map heat flow in dimension two.","PeriodicalId":45919,"journal":{"name":"Vietnam Journal of Mathematics","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-09-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"134886841","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-09-23DOI: 10.1007/s10013-023-00651-1
Ka Luen Cheung, Sen Wong
{"title":"On Classical Solutions of the Compressible Euler Equations for Generalized Chaplygin Gas with Qualitative Analysis","authors":"Ka Luen Cheung, Sen Wong","doi":"10.1007/s10013-023-00651-1","DOIUrl":"https://doi.org/10.1007/s10013-023-00651-1","url":null,"abstract":"","PeriodicalId":45919,"journal":{"name":"Vietnam Journal of Mathematics","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-09-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135959719","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-09-07DOI: 10.1007/s10013-023-00649-9
B. Dinh, Hy Duc Manh, Tran Thi Huyen Thanh
{"title":"Extragradient Algorithms with Linesearches for Solving Nonmonotone Equilibrium Problems in Banach Spaces","authors":"B. Dinh, Hy Duc Manh, Tran Thi Huyen Thanh","doi":"10.1007/s10013-023-00649-9","DOIUrl":"https://doi.org/10.1007/s10013-023-00649-9","url":null,"abstract":"","PeriodicalId":45919,"journal":{"name":"Vietnam Journal of Mathematics","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2023-09-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45914547","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-08-22DOI: 10.1007/s10013-023-00647-x
G. Misiołek, Xuan-Truong Vu
{"title":"On Continuity Properties of Solution Maps of the Generalized SQG Family","authors":"G. Misiołek, Xuan-Truong Vu","doi":"10.1007/s10013-023-00647-x","DOIUrl":"https://doi.org/10.1007/s10013-023-00647-x","url":null,"abstract":"","PeriodicalId":45919,"journal":{"name":"Vietnam Journal of Mathematics","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2023-08-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44350699","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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