Pub Date : 2026-01-29DOI: 10.1016/j.na.2026.114062
Jing Li
The purpose of this paper is to investigate the vanishing theorems problem of a generalized harmonic map in a class of sub-Riemannian manifolds. More specifically, we consider a horizontal functional related to the pullback metrics and introduce the concept of (weakly) stationary maps with potential H into Heisenberg groups. By using the stress-energy tensor method, we achieve some vanishing theorems for these maps under diverse proper conditions respectively.
{"title":"Vanishing theorems for F−CC stationary maps with potential into Heisenberg groups","authors":"Jing Li","doi":"10.1016/j.na.2026.114062","DOIUrl":"10.1016/j.na.2026.114062","url":null,"abstract":"<div><div>The purpose of this paper is to investigate the vanishing theorems problem of a generalized harmonic map in a class of sub-Riemannian manifolds. More specifically, we consider a horizontal functional <span><math><msubsup><mstyle><mi>Φ</mi></mstyle><mrow><mi>H</mi><mo>,</mo><mi>H</mi><mo>,</mo><mi>D</mi></mrow><mi>F</mi></msubsup></math></span> related to the pullback metrics and introduce the concept of (weakly) <span><math><mrow><mi>F</mi><mspace></mspace><mo>−</mo><mspace></mspace><mi>C</mi><mi>C</mi></mrow></math></span> stationary maps with potential <em>H</em> into Heisenberg groups. By using the stress-energy tensor method, we achieve some vanishing theorems for these maps under diverse proper conditions respectively.</div></div>","PeriodicalId":49749,"journal":{"name":"Nonlinear Analysis-Theory Methods & Applications","volume":"268 ","pages":"Article 114062"},"PeriodicalIF":1.3,"publicationDate":"2026-01-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146070768","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}
Pub Date : 2026-01-29DOI: 10.1016/j.na.2026.114060
Jonas Stange
We study a bulk-surface Cahn–Hilliard model with non-degenerate mobility and singular potentials in two dimensions. Following the ideas of the recent work by Conti, Galimberti, Gatti, and Giorgini [Calc. Var. Partial Differential Equations, 64(3):Paper No. 87, 32, 2025] for the Cahn–Hilliard equation with homogeneous Neumann boundary conditions, we show the uniqueness of weak solutions together with a continuous dependence estimate for sufficiently regular mobility functions. Next, under weaker assumptions on the mobility functions, we show the existence of a weak solution that exhibits the propagation of uniform-in-time regularity and satisfies the instantaneous separation property. Lastly, we consider the long-time behavior and prove that the unique weak solution converges to a solution of the stationary bulk-surface Cahn–Hilliard equation. Our approach for the uniqueness proof relies on a new well-posedness and regularity theory for a bulk-surface elliptic system with non-constant coefficients, which may be of independent interest.
{"title":"Well-posedness and long-time behavior of a bulk-surface Cahn–Hilliard model with non-degenerate mobility","authors":"Jonas Stange","doi":"10.1016/j.na.2026.114060","DOIUrl":"10.1016/j.na.2026.114060","url":null,"abstract":"<div><div>We study a bulk-surface Cahn–Hilliard model with non-degenerate mobility and singular potentials in two dimensions. Following the ideas of the recent work by Conti, Galimberti, Gatti, and Giorgini [Calc. Var. Partial Differential Equations, 64(3):Paper No. 87, 32, 2025] for the Cahn–Hilliard equation with homogeneous Neumann boundary conditions, we show the uniqueness of weak solutions together with a continuous dependence estimate for sufficiently regular mobility functions. Next, under weaker assumptions on the mobility functions, we show the existence of a weak solution that exhibits the propagation of uniform-in-time regularity and satisfies the instantaneous separation property. Lastly, we consider the long-time behavior and prove that the unique weak solution converges to a solution of the stationary bulk-surface Cahn–Hilliard equation. Our approach for the uniqueness proof relies on a new well-posedness and regularity theory for a bulk-surface elliptic system with non-constant coefficients, which may be of independent interest.</div></div>","PeriodicalId":49749,"journal":{"name":"Nonlinear Analysis-Theory Methods & Applications","volume":"268 ","pages":"Article 114060"},"PeriodicalIF":1.3,"publicationDate":"2026-01-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146070767","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}
Pub Date : 2026-01-28DOI: 10.1016/j.na.2026.114063
Geunsu Choi , Mingu Jung , Han Ju Lee , Óscar Roldán
We solve two main questions on linear structures of (non-)norm-attaining Lipschitz functions. First, we show that for every infinite metric space M, the set consisting of Lipschitz functions on M which do not strongly attain their norm and the zero function contains an isometric copy of ℓ∞, and moreover, those functions can be chosen not to attain their norm as functionals on the Lipschitz-free space over M. Second, we prove that for every infinite metric space M, neither the set of strongly norm-attaining Lipschitz functions on M nor the union of its complement with zero is ever a linear space. Furthermore, we observe that the set consisting of Lipschitz functions which cannot be approximated by strongly norm-attaining ones and the zero element contains ℓ∞ isometrically in all the known cases. Some natural observations and spaceability results are also investigated for Lipschitz functions that attain their norm in one way but do not in another.
{"title":"Linear structures of norm-attaining Lipschitz functions and their complements","authors":"Geunsu Choi , Mingu Jung , Han Ju Lee , Óscar Roldán","doi":"10.1016/j.na.2026.114063","DOIUrl":"10.1016/j.na.2026.114063","url":null,"abstract":"<div><div>We solve two main questions on linear structures of (non-)norm-attaining Lipschitz functions. First, we show that for every infinite metric space <em>M</em>, the set consisting of Lipschitz functions on <em>M</em> which do not strongly attain their norm and the zero function contains an isometric copy of ℓ<sub>∞</sub>, and moreover, those functions can be chosen not to attain their norm as functionals on the Lipschitz-free space over <em>M</em>. Second, we prove that for every infinite metric space <em>M</em>, neither the set of strongly norm-attaining Lipschitz functions on <em>M</em> nor the union of its complement with zero is ever a linear space. Furthermore, we observe that the set consisting of Lipschitz functions which cannot be approximated by strongly norm-attaining ones and the zero element contains ℓ<sub>∞</sub> isometrically in all the known cases. Some natural observations and spaceability results are also investigated for Lipschitz functions that attain their norm in one way but do not in another.</div></div>","PeriodicalId":49749,"journal":{"name":"Nonlinear Analysis-Theory Methods & Applications","volume":"267 ","pages":"Article 114063"},"PeriodicalIF":1.3,"publicationDate":"2026-01-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146078532","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}
Pub Date : 2026-01-28DOI: 10.1016/j.na.2026.114065
Jie Li
<div><div>In this paper, we consider the cauchy problem for the stochastic regularized dispersive wave (SDW) equations forced by the Gaussian process<span><span><span><math><mrow><mrow><mo>{</mo><mtable><mtr><mtd><mrow><mrow><mo>(</mo><mi>I</mi><mo>+</mo><mi>L</mi><mo>)</mo></mrow><mi>d</mi><mi>u</mi><mo>+</mo><mrow><mo>(</mo><msub><mi>u</mi><mi>x</mi></msub><mo>+</mo><msub><mrow><mo>(</mo><mi>h</mi><mrow><mo>(</mo><mi>u</mi><mo>)</mo></mrow><mo>)</mo></mrow><mi>x</mi></msub><mo>)</mo></mrow><mi>d</mi><mi>t</mi><mo>=</mo><mstyle><mi>Φ</mi></mstyle><mi>d</mi><mi>W</mi><mo>,</mo><mspace></mspace><mrow><mo>(</mo><mi>x</mi><mo>,</mo><mi>t</mi><mo>)</mo></mrow><mo>∈</mo><mi>R</mi><mo>×</mo><msup><mi>R</mi><mo>+</mo></msup><mo>,</mo></mrow></mtd></mtr><mtr><mtd><mrow><mi>u</mi><mrow><mo>(</mo><mi>x</mi><mo>,</mo><mn>0</mn><mo>)</mo></mrow><mo>=</mo><msub><mi>u</mi><mn>0</mn></msub><mo>,</mo><mspace></mspace><mi>x</mi><mo>∈</mo><mi>R</mi><mo>,</mo></mrow></mtd></mtr></mtable></mrow><mspace></mspace><mspace></mspace><mspace></mspace><mrow><mo>(</mo><mn>0.1</mn><mo>)</mo></mrow></mrow></math></span></span></span>where <span><math><mrow><mi>u</mi><mo>=</mo><mi>u</mi><mo>(</mo><mi>x</mi><mo>,</mo><mi>t</mi><mo>)</mo></mrow></math></span> is a real-valued function and <em>W</em> is a two-parameter Gaussian white noise on <span><math><mrow><mi>R</mi><mo>×</mo><msup><mi>R</mi><mo>+</mo></msup></mrow></math></span>. <em>L</em> is a Fourier multiplier operator and has a real representation <em>θ</em>(<em>ξ</em>) under the Fourier action. <em>h</em> is a real-valued, smooth function of one real variable. Φ is a Hilbert-Schmidt operator. Local well-posedness of (0.1) is obtained for <em>H<sup>s</sup></em> initial data, almost surely. If <span><math><mrow><mi>θ</mi><mrow><mo>(</mo><mi>ξ</mi><mo>)</mo></mrow><mo>=</mo><msup><mrow><mo>|</mo><mi>ξ</mi><mo>|</mo></mrow><mi>r</mi></msup></mrow></math></span> with <em>r</em> > 1, <span><math><mrow><mi>s</mi><mo>≥</mo><mfrac><mi>r</mi><mn>2</mn></mfrac></mrow></math></span> and <span><math><mrow><mi>h</mi><mrow><mo>(</mo><mi>u</mi><mo>)</mo></mrow><mo>=</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><msup><mi>u</mi><mn>2</mn></msup></mrow></math></span>, global well-posedness of (0.1)is obtained for <em>H<sup>s</sup></em> initial data, almost surely. Moreover, this essay also shows this global solution <span><math><mrow><mi>u</mi><mo>∈</mo><msubsup><mi>L</mi><mi>F</mi><msup><mn>2</mn><mi>α</mi></msup></msubsup><mrow><mo>(</mo><mstyle><mi>Ω</mi></mstyle><mo>;</mo><mi>C</mi><mrow><mo>(</mo><mrow><mo>[</mo><mn>0</mn><mo>,</mo><msub><mi>T</mi><mn>0</mn></msub><mo>]</mo></mrow><mo>;</mo><msup><mi>H</mi><mi>s</mi></msup><mrow><mo>(</mo><mi>R</mi><mo>)</mo></mrow><mo>)</mo></mrow><mo>)</mo></mrow></mrow></math></span> for any <span><math><mrow><mi>s</mi><mo>≥</mo><mfrac><mi>r</mi><mn>2</mn></mfrac></mrow></math></span> (<em>r</em> > 1) and any <span><math><mrow><mi>α</mi><mo>∈</mo><msup><mi>Z</mi><mo>+</mo></msup></mrow></math></s
本文考虑高斯过程{(I+L)du+(ux+(h(u))x)dt=ΦdW,(x,t)∈R×R+,u(x,0)=u0,x∈R,（0.1）所迫随机正则化色散波（SDW）方程的柯西问题，其中u=u（x,t）是实值函数，W是R×R+上的双参数高斯白噪声。L是傅里叶乘数算子在傅里叶作用下有一个实数表示θ（ξ）H是一个单实变量的实值光滑函数。Φ是Hilbert-Schmidt算子。初始数据的局部适定性为（0.1），几乎可以肯定。如果θ(ξ)=|ξ|r, r >； 1,s≥r2, h(u)=12u2，则Hs初始数据的全局适定性为（0.1），几乎可以肯定。此外，本文还给出了对于任意s≥r2 （R >； 1）和任意α∈Z+的全局解u∈LF2α（Ω；C([0,T0];Hs(R))）。
{"title":"Cauchy problem for stochastic regularized nonlinear dispersive wave equations","authors":"Jie Li","doi":"10.1016/j.na.2026.114065","DOIUrl":"10.1016/j.na.2026.114065","url":null,"abstract":"<div><div>In this paper, we consider the cauchy problem for the stochastic regularized dispersive wave (SDW) equations forced by the Gaussian process<span><span><span><math><mrow><mrow><mo>{</mo><mtable><mtr><mtd><mrow><mrow><mo>(</mo><mi>I</mi><mo>+</mo><mi>L</mi><mo>)</mo></mrow><mi>d</mi><mi>u</mi><mo>+</mo><mrow><mo>(</mo><msub><mi>u</mi><mi>x</mi></msub><mo>+</mo><msub><mrow><mo>(</mo><mi>h</mi><mrow><mo>(</mo><mi>u</mi><mo>)</mo></mrow><mo>)</mo></mrow><mi>x</mi></msub><mo>)</mo></mrow><mi>d</mi><mi>t</mi><mo>=</mo><mstyle><mi>Φ</mi></mstyle><mi>d</mi><mi>W</mi><mo>,</mo><mspace></mspace><mrow><mo>(</mo><mi>x</mi><mo>,</mo><mi>t</mi><mo>)</mo></mrow><mo>∈</mo><mi>R</mi><mo>×</mo><msup><mi>R</mi><mo>+</mo></msup><mo>,</mo></mrow></mtd></mtr><mtr><mtd><mrow><mi>u</mi><mrow><mo>(</mo><mi>x</mi><mo>,</mo><mn>0</mn><mo>)</mo></mrow><mo>=</mo><msub><mi>u</mi><mn>0</mn></msub><mo>,</mo><mspace></mspace><mi>x</mi><mo>∈</mo><mi>R</mi><mo>,</mo></mrow></mtd></mtr></mtable></mrow><mspace></mspace><mspace></mspace><mspace></mspace><mrow><mo>(</mo><mn>0.1</mn><mo>)</mo></mrow></mrow></math></span></span></span>where <span><math><mrow><mi>u</mi><mo>=</mo><mi>u</mi><mo>(</mo><mi>x</mi><mo>,</mo><mi>t</mi><mo>)</mo></mrow></math></span> is a real-valued function and <em>W</em> is a two-parameter Gaussian white noise on <span><math><mrow><mi>R</mi><mo>×</mo><msup><mi>R</mi><mo>+</mo></msup></mrow></math></span>. <em>L</em> is a Fourier multiplier operator and has a real representation <em>θ</em>(<em>ξ</em>) under the Fourier action. <em>h</em> is a real-valued, smooth function of one real variable. Φ is a Hilbert-Schmidt operator. Local well-posedness of (0.1) is obtained for <em>H<sup>s</sup></em> initial data, almost surely. If <span><math><mrow><mi>θ</mi><mrow><mo>(</mo><mi>ξ</mi><mo>)</mo></mrow><mo>=</mo><msup><mrow><mo>|</mo><mi>ξ</mi><mo>|</mo></mrow><mi>r</mi></msup></mrow></math></span> with <em>r</em> > 1, <span><math><mrow><mi>s</mi><mo>≥</mo><mfrac><mi>r</mi><mn>2</mn></mfrac></mrow></math></span> and <span><math><mrow><mi>h</mi><mrow><mo>(</mo><mi>u</mi><mo>)</mo></mrow><mo>=</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><msup><mi>u</mi><mn>2</mn></msup></mrow></math></span>, global well-posedness of (0.1)is obtained for <em>H<sup>s</sup></em> initial data, almost surely. Moreover, this essay also shows this global solution <span><math><mrow><mi>u</mi><mo>∈</mo><msubsup><mi>L</mi><mi>F</mi><msup><mn>2</mn><mi>α</mi></msup></msubsup><mrow><mo>(</mo><mstyle><mi>Ω</mi></mstyle><mo>;</mo><mi>C</mi><mrow><mo>(</mo><mrow><mo>[</mo><mn>0</mn><mo>,</mo><msub><mi>T</mi><mn>0</mn></msub><mo>]</mo></mrow><mo>;</mo><msup><mi>H</mi><mi>s</mi></msup><mrow><mo>(</mo><mi>R</mi><mo>)</mo></mrow><mo>)</mo></mrow><mo>)</mo></mrow></mrow></math></span> for any <span><math><mrow><mi>s</mi><mo>≥</mo><mfrac><mi>r</mi><mn>2</mn></mfrac></mrow></math></span> (<em>r</em> > 1) and any <span><math><mrow><mi>α</mi><mo>∈</mo><msup><mi>Z</mi><mo>+</mo></msup></mrow></math></s","PeriodicalId":49749,"journal":{"name":"Nonlinear Analysis-Theory Methods & Applications","volume":"267 ","pages":"Article 114065"},"PeriodicalIF":1.3,"publicationDate":"2026-01-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146078440","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}
Pub Date : 2026-01-27DOI: 10.1016/j.na.2026.114064
Caihong Chang , Zhengce Zhang
In this paper we prove two properties of positive weak solutions for quasilinear elliptic equation of the type , with f satisfying certain structure conditions and involving the product of the function and its gradient. First, we establish a priori estimates for all solutions by utilizing the well-known doubling lemma. Then, we use topological degree to prove the existence of positive weak solutions. Our proof is based on a priori bounds, which will be achieved by applying a blow-up technique developed in [Rev. Mat. Iberoam. 34 (2018) 195–220]. Since the gradient of solution is singular near the boundary, we adopt a suitable weighted norm that involves the distance function to describe this singularity, and then add the restrictions on the exponents of quasilinear equations to the exponent of the weight terms, thereby extending the assumptions regarding upper bounds on exponent of solution from Serrin exponent presented in [Nonlinear Anal. 220 (2020) 112873] to Sobolev exponent.
{"title":"A priori estimates and existence of solutions for quasilinear elliptic equations with nonlinear gradient terms","authors":"Caihong Chang , Zhengce Zhang","doi":"10.1016/j.na.2026.114064","DOIUrl":"10.1016/j.na.2026.114064","url":null,"abstract":"<div><div>In this paper we prove two properties of positive weak solutions for quasilinear elliptic equation of the type <span><math><mrow><mo>−</mo><msub><mstyle><mi>Δ</mi></mstyle><mi>m</mi></msub><mi>u</mi><mo>=</mo><mi>f</mi><mrow><mo>(</mo><mi>x</mi><mo>,</mo><mi>u</mi><mo>,</mo><mi>∇</mi><mi>u</mi><mo>)</mo></mrow></mrow></math></span>, with <em>f</em> satisfying certain structure conditions and involving the product of the function and its gradient. First, we establish a priori estimates for all solutions by utilizing the well-known doubling lemma. Then, we use topological degree to prove the existence of positive weak solutions. Our proof is based on a priori bounds, which will be achieved by applying a blow-up technique developed in [Rev. Mat. Iberoam. 34 (2018) 195–220]. Since the gradient of solution is singular near the boundary, we adopt a suitable weighted norm that involves the distance function to describe this singularity, and then add the restrictions on the exponents of quasilinear equations to the exponent of the weight terms, thereby extending the assumptions regarding upper bounds on exponent of solution from Serrin exponent presented in [Nonlinear Anal. 220 (2020) 112873] to Sobolev exponent.</div></div>","PeriodicalId":49749,"journal":{"name":"Nonlinear Analysis-Theory Methods & Applications","volume":"267 ","pages":"Article 114064"},"PeriodicalIF":1.3,"publicationDate":"2026-01-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146078438","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}
Pub Date : 2026-01-27DOI: 10.1016/j.na.2026.114061
Shokhrukh Y. Kholmatov , Paolo Piovano
The existence and the regularity results obtained in [41] for the variational model introduced in [40] to study the optimal shape of crystalline materials in the setting of stress-driven rearrangement instabilities (SDRI) are extended from two dimensions to any dimensions n ≥ 2. The energy is the sum of the elastic and the surface energy contributions, which cannot be decoupled, and depend on configurational pairs consisting of a set and a function that model the region occupied by the crystal and the bulk displacement field, respectively. By following the physical literature, the “driving stress” due to the mismatch between the ideal free-standing equilibrium lattice of the crystal with respect to adjacent materials is included in the model by considering a discontinuous mismatch strain in the elastic energy. Since two-dimensional methods and the methods used in the previous literature where Dirichlet boundary conditions instead of the mismatch strain and only the wetting regime were considered, cannot be employed in this setting, we proceed differently, by including in the analysis the dewetting regime and carefully analyzing the fine properties of energy-equibounded sequences. This analysis allows to establish both a compactness property in the family of admissible configurations and the lower semicontinuity of the energy with respect to the topology induced by the L1-convergence of sets and a.e. convergence of displacement fields, so that the direct method can be applied. We also prove that our arguments work as well in the setting with Dirichlet boundary conditions.
{"title":"Existence of minimizers for the SDRI model in Rn: Wetting and dewetting regimes with mismatch strain","authors":"Shokhrukh Y. Kholmatov , Paolo Piovano","doi":"10.1016/j.na.2026.114061","DOIUrl":"10.1016/j.na.2026.114061","url":null,"abstract":"<div><div>The existence and the regularity results obtained in [41] for the variational model introduced in [40] to study the optimal shape of crystalline materials in the setting of stress-driven rearrangement instabilities (SDRI) are extended from two dimensions to any dimensions <em>n</em> ≥ 2. The energy is the sum of the elastic and the surface energy contributions, which cannot be decoupled, and depend on configurational pairs consisting of a set and a function that model the region occupied by the crystal and the bulk displacement field, respectively. By following the physical literature, the “driving stress” due to the mismatch between the ideal free-standing equilibrium lattice of the crystal with respect to adjacent materials is included in the model by considering a discontinuous mismatch strain in the elastic energy. Since two-dimensional methods and the methods used in the previous literature where Dirichlet boundary conditions instead of the mismatch strain and only the wetting regime were considered, cannot be employed in this setting, we proceed differently, by including in the analysis the dewetting regime and carefully analyzing the fine properties of energy-equibounded sequences. This analysis allows to establish both a compactness property in the family of admissible configurations and the lower semicontinuity of the energy with respect to the topology induced by the <em>L</em><sup>1</sup>-convergence of sets and a.e. convergence of displacement fields, so that the direct method can be applied. We also prove that our arguments work as well in the setting with Dirichlet boundary conditions.</div></div>","PeriodicalId":49749,"journal":{"name":"Nonlinear Analysis-Theory Methods & Applications","volume":"267 ","pages":"Article 114061"},"PeriodicalIF":1.3,"publicationDate":"2026-01-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146078531","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}
Pub Date : 2026-01-21DOI: 10.1016/j.na.2026.114059
Claudianor O. Alves
In this work we use variational methods to prove the existence and concentration of nonnegative solutions for the following class of problemswhere Δ1 is the Laplacian operator, ϵ is a positive parameter, is a continuous function having a subcritical growth and is a continuous function with a local minimum.
{"title":"On existence of solutions to a class of problems involving the 1−Laplace operator in whole RN via penalization method","authors":"Claudianor O. Alves","doi":"10.1016/j.na.2026.114059","DOIUrl":"10.1016/j.na.2026.114059","url":null,"abstract":"<div><div>In this work we use variational methods to prove the existence and concentration of nonnegative solutions for the following class of problems<span><span><span><math><mrow><mo>−</mo><mi>ϵ</mi><msub><mstyle><mi>Δ</mi></mstyle><mn>1</mn></msub><mi>u</mi><mo>+</mo><mi>V</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mfrac><mi>u</mi><mrow><mo>|</mo><mi>u</mi><mo>|</mo></mrow></mfrac><mo>=</mo><mi>f</mi><mrow><mo>(</mo><mi>u</mi><mo>)</mo></mrow><mspace></mspace><mtext>in</mtext><mspace></mspace><msup><mi>R</mi><mi>N</mi></msup><mo>,</mo><mspace></mspace><mi>u</mi><mo>∈</mo><mi>B</mi><mi>V</mi><mrow><mo>(</mo><msup><mi>R</mi><mi>N</mi></msup><mo>)</mo></mrow><mo>,</mo></mrow></math></span></span></span>where Δ<sub>1</sub> is the <span><math><mrow><mn>1</mn><mo>−</mo></mrow></math></span>Laplacian operator, ϵ is a positive parameter, <span><math><mrow><mi>f</mi><mo>:</mo><mi>R</mi><mo>→</mo><mi>R</mi></mrow></math></span> is a continuous function having a subcritical growth and <span><math><mrow><mi>V</mi><mo>:</mo><msup><mrow><mi>R</mi></mrow><mi>N</mi></msup><mo>→</mo><mi>R</mi></mrow></math></span> is a continuous function with a local minimum.</div></div>","PeriodicalId":49749,"journal":{"name":"Nonlinear Analysis-Theory Methods & Applications","volume":"267 ","pages":"Article 114059"},"PeriodicalIF":1.3,"publicationDate":"2026-01-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146023412","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}
Pub Date : 2026-01-21DOI: 10.1016/j.na.2026.114058
Jordan Serres
We prove that every nearly spherical, positively curved surface is the contractive, volume-preserving image of a round sphere. The proof combines three main tools: the Ricci flow on surfaces, the Kim-Milman construction, and a multiscale Bakry-Émery criterion.
{"title":"Contractive transport maps from S2 to nearly spherical surfaces with positive Ricci curvature","authors":"Jordan Serres","doi":"10.1016/j.na.2026.114058","DOIUrl":"10.1016/j.na.2026.114058","url":null,"abstract":"<div><div>We prove that every nearly spherical, positively curved surface is the contractive, volume-preserving image of a round sphere. The proof combines three main tools: the Ricci flow on surfaces, the Kim-Milman construction, and a multiscale Bakry-Émery criterion.</div></div>","PeriodicalId":49749,"journal":{"name":"Nonlinear Analysis-Theory Methods & Applications","volume":"267 ","pages":"Article 114058"},"PeriodicalIF":1.3,"publicationDate":"2026-01-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146023414","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}
Pub Date : 2026-01-20DOI: 10.1016/j.na.2026.114056
Paulo Henryque C. Silva
In this paper, we establish two p-eigenvalue pinching sphere theorems, for the p-Laplacian, p > 1. The first result states that if the first non-zero p-eigenvalue of a closed Riemannian n-manifold with sectional curvature KM ≥ 1 is sufficiently close to the first non-zero p-eigenvalue of then M is homeomorphic to . The second states that if the first non-zero p-eigenvalue of a closed Riemannian n-manifold with Ricci curvature and injectivity radius injM ≥ i0 > 0 is sufficiently close to the first non-zero p-eigenvalue of then M is diffeomorphic to . Our results extend sphere theorems originally settled for the Laplacian by S. Croke [1] and G.P. Bessa [2] respectively.
{"title":"p-Eigenvalue pinching sphere theorems","authors":"Paulo Henryque C. Silva","doi":"10.1016/j.na.2026.114056","DOIUrl":"10.1016/j.na.2026.114056","url":null,"abstract":"<div><div>In this paper, we establish two <em>p</em>-eigenvalue pinching sphere theorems, for the <em>p</em>-Laplacian, <em>p</em> > 1. The first result states that if the first non-zero <em>p</em>-eigenvalue of a closed Riemannian <em>n</em>-manifold with sectional curvature <em>K<sub>M</sub></em> ≥ 1 is sufficiently close to the first non-zero <em>p</em>-eigenvalue of <span><math><msup><mi>S</mi><mi>n</mi></msup></math></span> then <em>M</em> is homeomorphic to <span><math><msup><mi>S</mi><mi>n</mi></msup></math></span>. The second states that if the first non-zero <em>p</em>-eigenvalue of a closed Riemannian <em>n</em>-manifold with Ricci curvature <span><math><mrow><msub><mrow><mrow><mi>R</mi></mrow><mi>i</mi><mi>c</mi></mrow><mi>M</mi></msub><mo>≥</mo><mrow><mo>(</mo><mi>n</mi><mo>−</mo><mn>1</mn><mo>)</mo></mrow></mrow></math></span> and injectivity radius inj<sub><em>M</em></sub> ≥ <em>i</em><sub>0</sub> > 0 is sufficiently close to the first non-zero <em>p</em>-eigenvalue of <span><math><msup><mi>S</mi><mi>n</mi></msup></math></span> then <em>M</em> is diffeomorphic to <span><math><msup><mi>S</mi><mi>n</mi></msup></math></span>. Our results extend sphere theorems originally settled for the Laplacian by S. Croke [1] and G.P. Bessa [2] respectively.</div></div>","PeriodicalId":49749,"journal":{"name":"Nonlinear Analysis-Theory Methods & Applications","volume":"267 ","pages":"Article 114056"},"PeriodicalIF":1.3,"publicationDate":"2026-01-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146023413","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}
Pub Date : 2026-01-16DOI: 10.1016/j.na.2026.114055
Kyungkeun Kang , Jörg Wolf
We study the partial regularity of suitable weak solutions for non-Newtonian Navier-Stokes equations that is specifically a power-law type of shear thickening flows. We prove a generalization of CKN theorem for the power in the range . As one of our main tools, we establish an ϵ-regularity criterion, that is, the smallness of a scaling invariant local norm for of the velocity filed, which seems to be of independent interest.
{"title":"On the partial regularity of suitable weak solutions to the equations of shear thickening fluids","authors":"Kyungkeun Kang , Jörg Wolf","doi":"10.1016/j.na.2026.114055","DOIUrl":"10.1016/j.na.2026.114055","url":null,"abstract":"<div><div>We study the partial regularity of suitable weak solutions for non-Newtonian Navier-Stokes equations that is specifically a power-law type of shear thickening flows. We prove a generalization of CKN theorem for the power in the range <span><math><mrow><mo>[</mo><mn>2</mn><mo>,</mo><mfrac><mn>11</mn><mn>5</mn></mfrac><mo>)</mo></mrow></math></span>. As one of our main tools, we establish an ϵ-regularity criterion, that is, the smallness of a scaling invariant local norm for <span><math><mrow><msubsup><mi>L</mi><mi>t</mi><mi>∞</mi></msubsup><msubsup><mi>L</mi><mi>x</mi><mn>2</mn></msubsup></mrow></math></span> of the velocity filed, which seems to be of independent interest.</div></div>","PeriodicalId":49749,"journal":{"name":"Nonlinear Analysis-Theory Methods & Applications","volume":"267 ","pages":"Article 114055"},"PeriodicalIF":1.3,"publicationDate":"2026-01-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145979448","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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