Calculus of Variations and Partial Differential Equations最新文献

{"title":"Best constants in subelliptic fractional Sobolev and Gagliardo-Nirenberg inequalities and ground states on stratified Lie groups.","authors":"Sekhar Ghosh, Vishvesh Kumar, Michael Ruzhansky","doi":"10.1007/s00526-025-03191-3","DOIUrl":"10.1007/s00526-025-03191-3","url":null,"abstract":"<p><p>In this paper, we establish the sharp fractional subelliptic Sobolev inequalities and Gagliardo-Nirenberg inequalities on stratified Lie groups. The best constants are given in terms of a ground state solution of a fractional subelliptic equation involving the fractional <i>p</i>-sublaplacian ( <math><mrow><mn>1</mn> <mo><</mo> <mi>p</mi> <mo><</mo> <mi>∞</mi></mrow> </math> ) on stratified Lie groups. We also prove the existence of ground state (least energy) solutions to nonlinear subelliptic fractional Schrödinger equation on stratified Lie groups. Different from the proofs of analogous results in the setting of classical Sobolev spaces on Euclidean spaces given by Weinstein (Comm. Math. Phys. 87(4):576-676, 1982/1983) using the rearrangement inequality which is not available in stratified Lie groups, we apply a subelliptic version of vanishing lemma due to Lions extended in the setting of stratified Lie groups combining it with the compact embedding theorem for subelliptic fractional Sobolev spaces obtained in our previous paper (Math. Ann. 388(4):4201-4249, 2024). We also present subelliptic fractional logarithmic Sobolev inequalities with explicit constants on stratified Lie groups. The main results are new for <math><mrow><mi>p</mi> <mo>=</mo> <mn>2</mn></mrow> </math> even in the context of the Heisenberg group.</p>","PeriodicalId":9478,"journal":{"name":"Calculus of Variations and Partial Differential Equations","volume":"65 1","pages":"28"},"PeriodicalIF":2.0,"publicationDate":"2026-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.ncbi.nlm.nih.gov/pmc/articles/PMC12701863/pdf/","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145762226","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Long-time behaviour and bifurcation analysis of a two-species aggregation-diffusion system on the torus.","authors":"José A Carrillo, Yurij Salmaniw","doi":"10.1007/s00526-025-03132-0","DOIUrl":"10.1007/s00526-025-03132-0","url":null,"abstract":"<p><p>We investigate stationary states, including their existence and stability, in a class of nonlocal aggregation-diffusion equations with linear diffusion and symmetric nonlocal interactions. For the scalar case, we extend previous results by showing that key model features, such as existence, regularity, bifurcation structure, and stability exchange, continue to hold under a mere bounded variation hypothesis. For the corresponding two-species system, we carry out a fully rigorous bifurcation analysis using the bifurcation theory of Crandall & Rabinowitz. This framework allows us to classify all solution branches from homogeneous states, with particular attention given to those arising from the self-interaction strength and the cross-interaction strength, as well as the stability of the branch at a point of critical stability. The analysis relies on an equivalent classification of solutions through fixed points of a nonlinear map, followed by a careful derivation of Fréchet derivatives up to third order. An interesting application to cell-cell adhesion arises from our analysis, yielding stable segregation patterns that appear at the onset of cell sorting in a modelling regime where all interactions are purely attractive.</p>","PeriodicalId":9478,"journal":{"name":"Calculus of Variations and Partial Differential Equations","volume":"65 1","pages":"19"},"PeriodicalIF":2.0,"publicationDate":"2026-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.ncbi.nlm.nih.gov/pmc/articles/PMC12681474/pdf/","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145707398","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Variational structures for the Fokker-Planck equation with general Dirichlet boundary conditions.","authors":"Filippo Quattrocchi","doi":"10.1007/s00526-025-03193-1","DOIUrl":"https://doi.org/10.1007/s00526-025-03193-1","url":null,"abstract":"<p><p>We prove the convergence of a modified Jordan-Kinderlehrer-Otto scheme to a solution to the Fokker-Planck equation in  <math><mrow><mi>Ω</mi> <mo>⋐</mo> <msup><mrow><mi>R</mi></mrow> <mi>d</mi></msup> </mrow> </math> with general-strictly positive and temporally constant-Dirichlet boundary conditions. We work under mild assumptions on the domain, the drift, and the initial datum. In the special case where  <math><mi>Ω</mi></math> is an interval in  <math> <msup><mrow><mi>R</mi></mrow> <mn>1</mn></msup> </math> , we prove that such a solution is a gradient flow-curve of maximal slope-within a suitable space of measures, endowed with a modified Wasserstein distance. Our discrete scheme and modified distance draw inspiration from contributions by A. Figalli and N. Gigli [J. Math. Pures Appl. 94, (2010), pp. 107-130], and J. Morales [J. Math. Pures Appl. 112, (2018), pp. 41-88] on an optimal-transport approach to evolution equations with Dirichlet boundary conditions. Similarly to these works, we allow the mass to flow from/to the boundary  <math><mrow><mi>∂</mi> <mi>Ω</mi></mrow> </math> throughout the evolution. However, our leading idea is to also keep track of the mass at the boundary by working with measures defined on the whole closure  <math><mover><mi>Ω</mi> <mo>¯</mo></mover> </math> . The driving functional is a modification of the classical relative entropy that also makes use of the information at the boundary. As an intermediate result, when  <math><mi>Ω</mi></math> is an interval in  <math> <msup><mrow><mi>R</mi></mrow> <mn>1</mn></msup> </math> , we find a formula for the descending slope of this geodesically nonconvex functional.</p>","PeriodicalId":9478,"journal":{"name":"Calculus of Variations and Partial Differential Equations","volume":"65 1","pages":"23"},"PeriodicalIF":2.0,"publicationDate":"2026-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.ncbi.nlm.nih.gov/pmc/articles/PMC12681467/pdf/","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145707396","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"JKO schemes with general transport costs.","authors":"Cale Rankin, Ting-Kam Leonard Wong","doi":"10.1007/s00526-026-03256-x","DOIUrl":"https://doi.org/10.1007/s00526-026-03256-x","url":null,"abstract":"<p><p>We modify the JKO scheme, which is a time discretization of the Wasserstein gradient flow, by replacing the Wasserstein distance with more general transport costs on manifolds. We show when the cost function has a mixed Hessian which defines a Riemannian metric, our modified JKO scheme converges, under suitable conditions, to the corresponding Riemannian Fokker-Planck equation. Thus on a Riemannian manifold one may replace the (squared) Riemannian distance with any cost function which induces the metric. Of interest is when the Riemannian distance is computationally intractable, but a suitable cost has a simple analytic expression. We consider the Fokker-Planck equation on compact submanifolds with the Neumann boundary condition and on complete Riemannian manifolds with a finite drift condition. As an application we consider Hessian manifolds, taking as a cost the Bregman divergence.</p>","PeriodicalId":9478,"journal":{"name":"Calculus of Variations and Partial Differential Equations","volume":"65 3","pages":"94"},"PeriodicalIF":2.0,"publicationDate":"2026-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.ncbi.nlm.nih.gov/pmc/articles/PMC12913252/pdf/","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146225655","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Thermo-elastodynamics of nonlinearly viscous solids.","authors":"Stefano Almi, Rufat Badal, Manuel Friedrich, Sebastian Schwarzacher","doi":"10.1007/s00526-026-03282-9","DOIUrl":"https://doi.org/10.1007/s00526-026-03282-9","url":null,"abstract":"<p><p>In this paper, we study the thermo-elastodynamics of nonlinearly viscous solids in the Kelvin-Voigt rheology where both the elastic and the viscous stress tensors comply with the frame-indifference principle. The system features a force balance including inertia in the frame of nonsimple materials and a heat-transfer equation which is governed by the Fourier law in the deformed configuration. Combining a staggered minimizing movement scheme for quasi-static thermoviscoelasticity [2, 37] with a variational approach to hyperbolic PDEs developed in [5], our main result consists in establishing the existence of weak solutions in the dynamic case. This is first achieved by including an additional higher-order regularization for the dissipation. Afterwards, this regularization can be removed by passing to a weaker formulation of the heat-transfer equation which complies with a total energy balance. The latter description hinges on regularity theory for the fourth order <i>p</i>-Laplacian which induces regularity estimates of the deformation beyond the standard estimates available from energy bounds. Besides being crucial for the proof, these extra regularity properties might be of independent interest and seem to be new in the setting of nonlinear viscoelasticity, also in the static or quasi-static case.</p>","PeriodicalId":9478,"journal":{"name":"Calculus of Variations and Partial Differential Equations","volume":"65 4","pages":"111"},"PeriodicalIF":2.0,"publicationDate":"2026-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.ncbi.nlm.nih.gov/pmc/articles/PMC12950105/pdf/","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"147343738","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Surgery and positive Bakry-Émery Ricci curvature.","authors":"Philipp Reiser, Francesca Tripaldi","doi":"10.1007/s00526-025-03211-2","DOIUrl":"https://doi.org/10.1007/s00526-025-03211-2","url":null,"abstract":"<p><p>We consider the problem of preserving weighted Riemannian metrics of positive Bakry-Émery Ricci curvature along surgery. We establish two theorems of this type: One for connected sums, and one for surgeries along higher-dimensional spheres. In contrast to known surgery results for positive Ricci curvature, these results are local, i.e. we only impose assumptions on the weighted metric locally around the sphere along which the surgery is performed. As application we then show that all closed, simply-connected spin 5-manifolds admit a weighted Riemannian metric of positive Bakry-Émery Ricci curvature. By a result of Lott, this also provides new examples of manifolds with a Riemannian metric of positive Ricci curvature.</p>","PeriodicalId":9478,"journal":{"name":"Calculus of Variations and Partial Differential Equations","volume":"65 2","pages":"38"},"PeriodicalIF":2.0,"publicationDate":"2026-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.ncbi.nlm.nih.gov/pmc/articles/PMC12795942/pdf/","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145970494","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"On De Giorgi's conjecture of nonlocal approximations for free-discontinuity problems: The symmetric gradient case.","authors":"S Almi, E Davoli, A Kubin, E Tasso","doi":"10.1007/s00526-026-03338-w","DOIUrl":"https://doi.org/10.1007/s00526-026-03338-w","url":null,"abstract":"<p><p>We prove that E. De Giorgi's conjecture for the nonlocal approximation of free-discontinuity problems extends to the case of functionals defined in terms of the symmetric gradient of the admissible field. After introducing a suitable class of continuous finite-difference approximants, we show the compactness of deformations with equibounded energies, as well as their Gamma-convergence. The compactness analysis is a crucial hurdle, which we overcome by generalizing a Fréchet-Kolmogorov approach previously introduced by two of the authors. A second essential difficulty is the identification of the limiting space of admissible deformations, since a control on the directional variations is, a priori, only available in average. A limiting representation in GSBD is eventually established via a novel characterization of this space.</p>","PeriodicalId":9478,"journal":{"name":"Calculus of Variations and Partial Differential Equations","volume":"65 5","pages":"160"},"PeriodicalIF":2.0,"publicationDate":"2026-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.ncbi.nlm.nih.gov/pmc/articles/PMC13070975/pdf/","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"147688520","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Parabolic-elliptic and indirect-direct simplifications in chemotaxis systems driven by indirect signalling.","authors":"Le Trong Thanh Bui, Thi Kim Loan Huynh, Bao Quoc Tang, Bao-Ngoc Tran","doi":"10.1007/s00526-025-03247-4","DOIUrl":"https://doi.org/10.1007/s00526-025-03247-4","url":null,"abstract":"&lt;p&gt;&lt;p&gt;Singular limits for the following indirect signalling chemotaxis system &lt;dispformula&gt; &lt;math&gt; &lt;mrow&gt; &lt;mtable&gt; &lt;mtr&gt; &lt;mtd&gt; &lt;mfenced&gt; &lt;mrow&gt; &lt;mtable&gt; &lt;mtr&gt; &lt;mtd&gt; &lt;mrow&gt;&lt;msub&gt;&lt;mi&gt;∂&lt;/mi&gt; &lt;mi&gt;t&lt;/mi&gt;&lt;/msub&gt; &lt;mi&gt;n&lt;/mi&gt; &lt;mo&gt;=&lt;/mo&gt; &lt;mi&gt;Δ&lt;/mi&gt; &lt;mi&gt;n&lt;/mi&gt; &lt;mo&gt;-&lt;/mo&gt; &lt;mi&gt;∇&lt;/mi&gt; &lt;mo&gt;·&lt;/mo&gt; &lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt; &lt;mi&gt;n&lt;/mi&gt; &lt;mi&gt;∇&lt;/mi&gt; &lt;mi&gt;c&lt;/mi&gt; &lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt; &lt;/mrow&gt; &lt;/mtd&gt; &lt;mtd&gt;&lt;mrow&gt;&lt;mtext&gt;in&lt;/mtext&gt; &lt;mspace&gt;&lt;/mspace&gt; &lt;mi&gt;Ω&lt;/mi&gt; &lt;mo&gt;×&lt;/mo&gt; &lt;mo&gt;(&lt;/mo&gt; &lt;mn&gt;0&lt;/mn&gt; &lt;mo&gt;,&lt;/mo&gt; &lt;mi&gt;∞&lt;/mi&gt; &lt;mo&gt;)&lt;/mo&gt; &lt;mo&gt;,&lt;/mo&gt;&lt;/mrow&gt; &lt;/mtd&gt; &lt;/mtr&gt; &lt;mtr&gt; &lt;mtd&gt;&lt;mrow&gt;&lt;mrow&gt;&lt;/mrow&gt; &lt;mi&gt;ε&lt;/mi&gt; &lt;msub&gt;&lt;mi&gt;∂&lt;/mi&gt; &lt;mi&gt;t&lt;/mi&gt;&lt;/msub&gt; &lt;mi&gt;c&lt;/mi&gt; &lt;mo&gt;=&lt;/mo&gt; &lt;mi&gt;Δ&lt;/mi&gt; &lt;mi&gt;c&lt;/mi&gt; &lt;mo&gt;-&lt;/mo&gt; &lt;mi&gt;c&lt;/mi&gt; &lt;mo&gt;+&lt;/mo&gt; &lt;mi&gt;w&lt;/mi&gt;&lt;/mrow&gt; &lt;/mtd&gt; &lt;mtd&gt;&lt;mrow&gt;&lt;mtext&gt;in&lt;/mtext&gt; &lt;mspace&gt;&lt;/mspace&gt; &lt;mi&gt;Ω&lt;/mi&gt; &lt;mo&gt;×&lt;/mo&gt; &lt;mo&gt;(&lt;/mo&gt; &lt;mn&gt;0&lt;/mn&gt; &lt;mo&gt;,&lt;/mo&gt; &lt;mi&gt;∞&lt;/mi&gt; &lt;mo&gt;)&lt;/mo&gt; &lt;mo&gt;,&lt;/mo&gt;&lt;/mrow&gt; &lt;/mtd&gt; &lt;/mtr&gt; &lt;mtr&gt; &lt;mtd&gt;&lt;mrow&gt;&lt;mrow&gt;&lt;/mrow&gt; &lt;mi&gt;ε&lt;/mi&gt; &lt;msub&gt;&lt;mi&gt;∂&lt;/mi&gt; &lt;mi&gt;t&lt;/mi&gt;&lt;/msub&gt; &lt;mi&gt;w&lt;/mi&gt; &lt;mo&gt;=&lt;/mo&gt; &lt;mi&gt;τ&lt;/mi&gt; &lt;mi&gt;Δ&lt;/mi&gt; &lt;mi&gt;w&lt;/mi&gt; &lt;mo&gt;-&lt;/mo&gt; &lt;mi&gt;w&lt;/mi&gt; &lt;mo&gt;+&lt;/mo&gt; &lt;mi&gt;n&lt;/mi&gt;&lt;/mrow&gt; &lt;/mtd&gt; &lt;mtd&gt;&lt;mrow&gt;&lt;mtext&gt;in&lt;/mtext&gt; &lt;mspace&gt;&lt;/mspace&gt; &lt;mi&gt;Ω&lt;/mi&gt; &lt;mo&gt;×&lt;/mo&gt; &lt;mo&gt;(&lt;/mo&gt; &lt;mn&gt;0&lt;/mn&gt; &lt;mo&gt;,&lt;/mo&gt; &lt;mi&gt;∞&lt;/mi&gt; &lt;mo&gt;)&lt;/mo&gt; &lt;mo&gt;,&lt;/mo&gt;&lt;/mrow&gt; &lt;/mtd&gt; &lt;/mtr&gt; &lt;mtr&gt; &lt;mtd&gt;&lt;mrow&gt;&lt;mrow&gt;&lt;/mrow&gt; &lt;msub&gt;&lt;mi&gt;∂&lt;/mi&gt; &lt;mi&gt;ν&lt;/mi&gt;&lt;/msub&gt; &lt;mi&gt;n&lt;/mi&gt; &lt;mo&gt;=&lt;/mo&gt; &lt;msub&gt;&lt;mi&gt;∂&lt;/mi&gt; &lt;mi&gt;ν&lt;/mi&gt;&lt;/msub&gt; &lt;mi&gt;c&lt;/mi&gt; &lt;mo&gt;=&lt;/mo&gt; &lt;msub&gt;&lt;mi&gt;∂&lt;/mi&gt; &lt;mi&gt;ν&lt;/mi&gt;&lt;/msub&gt; &lt;mi&gt;w&lt;/mi&gt; &lt;mo&gt;=&lt;/mo&gt; &lt;mn&gt;0&lt;/mn&gt; &lt;mo&gt;,&lt;/mo&gt;&lt;/mrow&gt; &lt;/mtd&gt; &lt;mtd&gt;&lt;mrow&gt;&lt;mtext&gt;on&lt;/mtext&gt; &lt;mspace&gt;&lt;/mspace&gt; &lt;mi&gt;∂&lt;/mi&gt; &lt;mi&gt;Ω&lt;/mi&gt; &lt;mo&gt;×&lt;/mo&gt; &lt;mo&gt;(&lt;/mo&gt; &lt;mn&gt;0&lt;/mn&gt; &lt;mo&gt;,&lt;/mo&gt; &lt;mi&gt;∞&lt;/mi&gt; &lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt; &lt;/mtd&gt; &lt;/mtr&gt; &lt;/mtable&gt; &lt;/mrow&gt; &lt;/mfenced&gt; &lt;/mtd&gt; &lt;/mtr&gt; &lt;/mtable&gt; &lt;/mrow&gt; &lt;/math&gt; &lt;/dispformula&gt; are investigated. More precisely, we study parabolic-elliptic simplification, or PES, &lt;math&gt;&lt;mrow&gt;&lt;mi&gt;ε&lt;/mi&gt; &lt;mo&gt;→&lt;/mo&gt; &lt;msup&gt;&lt;mn&gt;0&lt;/mn&gt; &lt;mo&gt;+&lt;/mo&gt;&lt;/msup&gt; &lt;/mrow&gt; &lt;/math&gt; with fixed &lt;math&gt;&lt;mrow&gt;&lt;mi&gt;τ&lt;/mi&gt; &lt;mo&gt;&gt;&lt;/mo&gt; &lt;mn&gt;0&lt;/mn&gt;&lt;/mrow&gt; &lt;/math&gt; up to the critical dimension &lt;math&gt;&lt;mrow&gt;&lt;mi&gt;N&lt;/mi&gt; &lt;mo&gt;=&lt;/mo&gt; &lt;mn&gt;4&lt;/mn&gt;&lt;/mrow&gt; &lt;/math&gt; , and indirect-direct simplification, or IDS, &lt;math&gt; &lt;mrow&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt; &lt;mi&gt;ε&lt;/mi&gt; &lt;mo&gt;,&lt;/mo&gt; &lt;mi&gt;τ&lt;/mi&gt; &lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt; &lt;mo&gt;→&lt;/mo&gt; &lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt; &lt;msup&gt;&lt;mn&gt;0&lt;/mn&gt; &lt;mo&gt;+&lt;/mo&gt;&lt;/msup&gt; &lt;mo&gt;,&lt;/mo&gt; &lt;msup&gt;&lt;mn&gt;0&lt;/mn&gt; &lt;mo&gt;+&lt;/mo&gt;&lt;/msup&gt; &lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt; &lt;/mrow&gt; &lt;/math&gt; up to the critical dimension &lt;math&gt;&lt;mrow&gt;&lt;mi&gt;N&lt;/mi&gt; &lt;mo&gt;=&lt;/mo&gt; &lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt; &lt;/math&gt; . These are relevant in biological situations where the signalling process is on a much faster time scale compared to the species diffusion and all interactions. Showing singular limits in critical dimensions is challenging. To deal with the PES, we carefully combine the entropy function, an Adam-type inequality, the regularisation of slow evolution, and an energy equation method to obtain strong convergence in representative spaces. For the IDS, a bootstrap argument concerning the &lt;math&gt;&lt;msup&gt;&lt;mi&gt;L&lt;/mi&gt; &lt;mi&gt;p&lt;/mi&gt;&lt;/msup&gt; &lt;/math&gt; -energy function is devised, which allows us to obtain suitable uniform bounds for the singul","PeriodicalId":9478,"journal":{"name":"Calculus of Variations and Partial Differential Equations","volume":"65 3","pages":"76"},"PeriodicalIF":2.0,"publicationDate":"2026-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.ncbi.nlm.nih.gov/pmc/articles/PMC12913282/pdf/","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146225681","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"<ArticleTitle xmlns:ns0=\"http://www.w3.org/1998/Math/MathML\">Energy identity and no neck property for <ns0:math><ns0:mi>ε</ns0:mi></ns0:math> -harmonic and <ns0:math><ns0:mi>α</ns0:mi></ns0:math> -harmonic maps into homogeneous target manifolds.","authors":"Carolin Bayer, Andrew M Roberts","doi":"10.1007/s00526-025-03233-w","DOIUrl":"10.1007/s00526-025-03233-w","url":null,"abstract":"<p><p>In this paper we show the energy identity and the no-neck property for <math><mi>ε</mi></math> - and <math><mi>α</mi></math> -harmonic maps with homogeneous target manifolds. To prove this in the <math><mi>ε</mi></math> -harmonic case we introduce the idea of using an equivariant embedding of the homogeneous target manifold.</p>","PeriodicalId":9478,"journal":{"name":"Calculus of Variations and Partial Differential Equations","volume":"65 2","pages":"57"},"PeriodicalIF":2.0,"publicationDate":"2026-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.ncbi.nlm.nih.gov/pmc/articles/PMC12795924/pdf/","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145970482","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Explicit minimisers for anisotropic Riesz energies.","authors":"R L Frank, J Mateu, M G Mora, L Rondi, L Scardia, J Verdera","doi":"10.1007/s00526-025-03185-1","DOIUrl":"https://doi.org/10.1007/s00526-025-03185-1","url":null,"abstract":"<p><p>In this paper we characterise the energy minimisers of a class of nonlocal interaction energies where the attraction is quadratic, and the repulsion is Riesz-like and anisotropic. In particular we show that, if the Fourier transform of the repulsive potential is positive, the minimiser is supported on a fully-dimensional ellipsoid, and its density is given by a Barenblatt-type profile. Our technique of proof is based on a Fourier representation of the potential of such measures, that extends a previous formula established by some of the authors in the Coulomb case.</p>","PeriodicalId":9478,"journal":{"name":"Calculus of Variations and Partial Differential Equations","volume":"65 1","pages":"34"},"PeriodicalIF":2.0,"publicationDate":"2026-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.ncbi.nlm.nih.gov/pmc/articles/PMC12701851/pdf/","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145762323","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}