Calculus of Variations and Partial Differential Equations最新文献

{"title":"Schauder estimates for parabolic equations with degenerate or singular weights","authors":"Alessandro Audrito, Gabriele Fioravanti, Stefano Vita","doi":"10.1007/s00526-024-02809-2","DOIUrl":"https://doi.org/10.1007/s00526-024-02809-2","url":null,"abstract":"<p>We establish some <span>(C^{0,alpha })</span> and <span>(C^{1,alpha })</span> regularity estimates for a class of weighted parabolic problems in divergence form. The main novelty is that the weights may vanish or explode on a characteristic hyperplane <span>(Sigma )</span> as a power <span>(a &gt; -1)</span> of the distance to <span>(Sigma )</span>. The estimates we obtain are sharp with respect to the assumptions on coefficients and data. Our methods rely on a regularization of the equation and some uniform regularity estimates combined with a Liouville theorem and an approximation argument. As a corollary of our main result, we obtain similar <span>(C^{1,alpha })</span> estimates when the degeneracy/singularity of the weight occurs on a regular hypersurface of cylindrical type.</p>","PeriodicalId":9478,"journal":{"name":"Calculus of Variations and Partial Differential Equations","volume":"75 1","pages":""},"PeriodicalIF":2.1,"publicationDate":"2024-08-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142192560","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}

{"title":"Interior Hölder estimate for the linearized complex Monge–Ampère equation","authors":"Yulun Xu","doi":"10.1007/s00526-024-02814-5","DOIUrl":"https://doi.org/10.1007/s00526-024-02814-5","url":null,"abstract":"<p>Let <span>(w_0)</span> be a bounded, <span>(C^3)</span>, strictly plurisubharmonic function defined on <span>(B_1subset mathbb {C}^n)</span>. Then <span>(w_0)</span> has a neighborhood in <span>(L^{infty }(B_1))</span>. Suppose that we have a function <span>(phi )</span> in this neighborhood with <span>(1-varepsilon le MA(phi )le 1+varepsilon )</span> and there exists a function <i>u</i> solving the linearized complex Monge–Amp<span>(grave{text {e}})</span>re equation: <span>(det(phi _{kbar{l}})phi ^{ibar{j}}u_{ibar{j}}=0)</span>. Then there exist constants <span>(alpha &gt;0)</span> and <i>C</i> such that <span>(|u|_{C^{alpha }(B_{frac{1}{2}}(0))}le C)</span>, where <span>(alpha &gt;0)</span> depends on <i>n</i> and <i>C</i> depends on <i>n</i> and <span>(|u|_{L^{infty }(B_1(0))})</span>, as long as <span>(epsilon )</span> is small depending on <i>n</i>. This partially generalizes Caffarelli–Gutierrez’s estimate for linearized real Monge–Amp<span>(grave{text {e}})</span>re equation to the complex version.\u0000</p>","PeriodicalId":9478,"journal":{"name":"Calculus of Variations and Partial Differential Equations","volume":"24 1","pages":""},"PeriodicalIF":2.1,"publicationDate":"2024-08-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142192557","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}

{"title":"Sharp Sobolev inequalities on noncompact Riemannian manifolds with $$textsf{Ric}ge 0$$ via optimal transport theory","authors":"Alexandru Kristály","doi":"10.1007/s00526-024-02810-9","DOIUrl":"https://doi.org/10.1007/s00526-024-02810-9","url":null,"abstract":"<p>In their seminal work, Cordero-Erausquin, Nazaret and Villani (Adv Math 182(2):307-332, 2004) proved sharp Sobolev inequalities in Euclidean spaces via <i>Optimal Transport</i>, raising the question whether their approach is powerful enough to produce sharp Sobolev inequalities also on Riemannian manifolds. By using <span>(L^1)</span>-optimal transport approach, the compact case has been successfully treated by Cavalletti and Mondino (Geom Topol 21:603-645, 2017), even on metric measure spaces verifying the synthetic lower Ricci curvature bound. In the present paper we affirmatively answer the above question for noncompact Riemannian manifolds with non-negative Ricci curvature; namely, by using Optimal Transport theory with quadratic distance cost, sharp <span>(L^p)</span>-Sobolev and <span>(L^p)</span>-logarithmic Sobolev inequalities (both for <span>(p&gt;1)</span> and <span>(p=1)</span>) are established, where the sharp constants contain the <i>asymptotic volume ratio</i> arising from precise asymptotic properties of the Talentian and Gaussian bubbles, respectively. As a byproduct, we give an alternative, elementary proof to the main result of do Carmo and Xia (Math 140:818-826, 2004) and subsequent results, concerning the quantitative volume non-collapsing estimates on Riemannian manifolds with non-negative Ricci curvature that support Sobolev inequalities.</p>","PeriodicalId":9478,"journal":{"name":"Calculus of Variations and Partial Differential Equations","volume":"75 1","pages":""},"PeriodicalIF":2.1,"publicationDate":"2024-08-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142192717","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}

{"title":"Hessian estimates for the Lagrangian mean curvature flow","authors":"Arunima Bhattacharya, Jeremy Wall","doi":"10.1007/s00526-024-02812-7","DOIUrl":"https://doi.org/10.1007/s00526-024-02812-7","url":null,"abstract":"<p>In this paper, we prove interior Hessian estimates for shrinkers, expanders, translators, and rotators of the Lagrangian mean curvature flow under the assumption that the Lagrangian phase is hypercritical. We further extend our results to a broader class of Lagrangian mean curvature type equations.</p>","PeriodicalId":9478,"journal":{"name":"Calculus of Variations and Partial Differential Equations","volume":"22 1","pages":""},"PeriodicalIF":2.1,"publicationDate":"2024-08-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142192561","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}

{"title":"Multiple solutions for (p, q)-Laplacian equations in $$mathbb {R}^N$$ with critical or subcritical exponents","authors":"Shibo Liu, Kanishka Perera","doi":"10.1007/s00526-024-02811-8","DOIUrl":"https://doi.org/10.1007/s00526-024-02811-8","url":null,"abstract":"<p>In this paper we study the following <span>(left( p,qright) )</span>-Laplacian equation with critical exponent </p><span>$$begin{aligned} -Delta _{p}u-Delta _{q}u=lambda h(x)|u|^{r-2}u+g(x)|u|^{p^{*} -2}u quad text {in }mathbb {R}^{N} , end{aligned}$$</span><p>where <span>(1&lt;qle p&lt;r&lt;p^{*})</span>. After establishing <span>((PS)_c)</span> condition for <span>(cin (0,c^*))</span> for a certain constant <span>(c^*)</span> by employing the concentration compactness principle of Lions, multiple solutions for <span>(lambda gg 1)</span> are obtained by applying a critical point theorem due to Perera (J Anal Math, 2023. arxiv:2308.07901). A similar problem with subcritical exponents is also considered.</p>","PeriodicalId":9478,"journal":{"name":"Calculus of Variations and Partial Differential Equations","volume":"71 1","pages":""},"PeriodicalIF":2.1,"publicationDate":"2024-08-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142192439","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}

{"title":"The effects of long-range interaction to wave propagation","authors":"Chao-Nien Chen, Yung-Sze Choi, Chih-Chiang Huang, Shyuh-yaur Tzeng","doi":"10.1007/s00526-024-02783-9","DOIUrl":"https://doi.org/10.1007/s00526-024-02783-9","url":null,"abstract":"<p>The mechanisms responsible for pattern formation have attracted a great deal of attention since Alan Turing elucidated his fascinating idea on diffusion-induced instability of steady states. Subsequent studies on the models demonstrated an entirely different class of solutions; namely localized structures composing of steadily moving fronts and pulses. In such energy-driven motion, the combination of short and long-range interaction plays an important ingredient for the generation of complex patterns. This competition on traveling wave dynamics, commonly observed in many physical and chemical phenomena, will be highlighted.\u0000</p>","PeriodicalId":9478,"journal":{"name":"Calculus of Variations and Partial Differential Equations","volume":"91 1","pages":""},"PeriodicalIF":2.1,"publicationDate":"2024-08-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141937539","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}

{"title":"Sharp decay estimates and asymptotic stability for incompressible MHD equations without viscosity or magnetic diffusion","authors":"Yaowei Xie, Quansen Jiu, Jitao Liu","doi":"10.1007/s00526-024-02799-1","DOIUrl":"https://doi.org/10.1007/s00526-024-02799-1","url":null,"abstract":"<p>Whether the global existence and uniqueness of strong solutions to <i>n</i>-dimensional incompressible magnetohydrodynamic (<i>MHD for short</i>) equations with only kinematic viscosity or magnetic diffusion holds true or not remains an outstanding open problem. In recent years, stared from the pioneer work by Lin and Zhang (Commun Pure Appl Math 67(4):531–580, 2014), much more attention has been paid to the case when the magnetic field close to an equilibrium state (<i>the background magnetic field for short</i>). Specifically, when the background magnetic field satisfies the Diophantine condition (see (1.2) for details), Chen et al. (Sci China Math 41:1–10, 2022) first studied the perturbation system and established the decay estimates and asymptotic stability of its solutions in 3D periodic domain <span>(mathbb {T}^3)</span>, which was then improved to <span>(H^{(3+2beta )r+5+(alpha +2beta )}(mathbb {T}^2))</span> for 2D periodic domain <span>(mathbb {T}^2)</span> and any <span>(alpha &gt;0)</span>, <span>(beta &gt;0)</span> by Zhai (J Differ Equ 374:267–278, 2023). In this paper, we seek to find the optimal decay estimates and improve the space where the global stability is taking place. Through deeply exploring and effectively utilizing the structure of perturbation system, we discover a <i>new</i> dissipative mechanism, which enables us to establish the decay estimates in the Sobolev spaces with <i>much lower</i> regularity. Based on the above discovery, we <i>greatly</i> reduce the initial regularity requirement of aforesaid two works from <span>(H^{4r+7}(mathbb {T}^3))</span> and <span>(H^{(3+2beta )r+5+(alpha +2beta )}(mathbb {T}^2))</span> to <span>(H^{(3r+3)^+}(mathbb {T}^n))</span> for <span>(r&gt;n-1)</span> when <span>(n=3)</span> and <span>(n=2)</span> respectively. Additionally, we first present the linear stability result via the method of spectral analysis in this paper. From which, the decay estimates obtained for the nonlinear system can be seen as <i>sharp</i> in the sense that they are in line with those for the linearized system.</p>","PeriodicalId":9478,"journal":{"name":"Calculus of Variations and Partial Differential Equations","volume":"162 1","pages":""},"PeriodicalIF":2.1,"publicationDate":"2024-08-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141937535","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}

{"title":"Convergence of critical points for a phase-field approximation of 1D cohesive fracture energies","authors":"Marco Bonacini, Flaviana Iurlano","doi":"10.1007/s00526-024-02786-6","DOIUrl":"https://doi.org/10.1007/s00526-024-02786-6","url":null,"abstract":"<p>Variational models for cohesive fracture are based on the idea that the fracture energy is released gradually as the crack opening grows. Recently, [21] proposed a variational approximation via <span>(Gamma )</span>-convergence of a class of cohesive fracture energies by phase-field energies of Ambrosio-Tortorelli type, which may be also used as regularization for numerical simulations. In this paper we address the question of the asymptotic behaviour of critical points of the phase-field energies in the one-dimensional setting: we show that they converge to a selected class of critical points of the limit functional. Conversely, each critical point in this class can be approximated by a family of critical points of the phase-field functionals.</p>","PeriodicalId":9478,"journal":{"name":"Calculus of Variations and Partial Differential Equations","volume":"48 1","pages":""},"PeriodicalIF":2.1,"publicationDate":"2024-08-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141937536","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}

{"title":"Riesz transform and Hardy spaces related to elliptic operators having Robin boundary conditions on Lipschitz domains with their applications to optimal endpoint regularity estimates","authors":"Dachun Yang, Sibei Yang, Yang Zou","doi":"10.1007/s00526-024-02785-7","DOIUrl":"https://doi.org/10.1007/s00526-024-02785-7","url":null,"abstract":"<p>Let <span>(nge 2)</span> and <span>(Omega )</span> be a bounded Lipschitz domain of <span>(mathbb {R}^n)</span>. Assume that <span>(L_R)</span> is a second-order divergence form elliptic operator having real-valued, bounded, symmetric, and measurable coefficients on <span>(L^2(Omega ))</span> with the Robin boundary condition. In this article, via first obtaining the Hölder estimate of the heat kernels of <span>(L_R)</span>, the authors establish a new atomic characterization of the Hardy space <span>(H^p_{L_R}(Omega ))</span> associated with <span>(L_R)</span>. Using this, the authors further show that, for any given <span>(pin (frac{n}{n+delta _0},1])</span>, </p><span>$$begin{aligned} H^p_z(Omega )+L^infty (Omega )=H^p_{L_N}(Omega )=H^p_{L_R}(Omega )subsetneqq H^p_{L_D}(Omega )=H^p_r(Omega ), end{aligned}$$</span><p>where <span>(H^p_{L_D}(Omega ))</span> and <span>(H^p_{L_N}(Omega ))</span> denote the Hardy spaces on <span>(Omega )</span> associated with the corresponding elliptic operators respectively having the Dirichlet and the Neumann boundary conditions, <span>(H^p_z(Omega ))</span> and <span>(H^p_r(Omega ))</span> respectively denote the “supported type” and the “restricted type” Hardy spaces on <span>(Omega )</span>, and <span>(delta _0in (0,1])</span> is the critical index depending on the operators <span>(L_D)</span>, <span>(L_N)</span>, and <span>(L_R)</span>. The authors then obtain the boundedness of the Riesz transform <span>(nabla L_R^{-1/2})</span> on the Lebesgue space <span>(L^{p}(Omega ))</span> when <span>(pin (1,infty ))</span> [if <span>(p&gt;2)</span>, some extra assumptions are needed] and its boundedness from <span>(H_{L_R}^{p}(Omega ))</span> to <span>(L^{p}(Omega ))</span> when <span>(pin (0,1])</span> or to <span>(H^{p}_r(Omega ))</span> when <span>(pin (frac{n}{n+1},1])</span>. As applications, the authors further obtain the global regularity estimates, in <span>(L^{p}(Omega ))</span> when <span>(pin (0,p_0))</span> and in <span>(H^{p}_r(Omega ))</span> when <span>(pin (frac{n}{n+1},1])</span>, for the inhomogeneous Robin problem of <span>(L_R)</span> on <span>(Omega )</span>, where <span>(p_0in (2,infty ))</span> is a constant depending only on <i>n</i>, <span>(Omega )</span>, and the operator <span>(L_R)</span>. The main novelties of these results are that the range <span>((0,p_0))</span> of <i>p</i> for the global regularity estimates in the scale of <span>(L^p(Omega ))</span> is sharp and that, in some sense, the space <span>(X{:}{=}H^1_{L_R}(Omega ))</span> is also optimal to guarantee both the boundedness of <span>(nabla L^{-1/2}_R)</span> from <i>X</i> to <span>(L^1(Omega ))</span> or to <span>(H^1_r(Omega ))</span> and the global regularity estimate <span>(Vert nabla uVert _{L^{frac{n}{n-1}} (Omega ;,mathbb {R}^n)}le CVert fVert _{X})</span> for inhomogeneous Robin problems with <i>C</i> being a positive constant independent of both <i>u</i> and <i>f</i>.</p>","PeriodicalId":9478,"journal":{"name":"Calculus of Variations and Partial Differential Equations","volume":"41 1","pages":""},"PeriodicalIF":2.1,"publicationDate":"2024-08-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141937537","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}

{"title":"Global stability of the Dirac–Klein–Gordon system in two and three space dimensions","authors":"Qian Zhang","doi":"10.1007/s00526-024-02803-8","DOIUrl":"https://doi.org/10.1007/s00526-024-02803-8","url":null,"abstract":"<p>In this paper we study global nonlinear stability for the Dirac–Klein–Gordon system in two and three space dimensions for small and regular initial data. In the case of two space dimensions, we consider the Dirac–Klein–Gordon system with a massless Dirac field and a massive scalar field, and prove global existence, sharp time decay estimates and linear scattering for the solutions. In the case of three space dimensions, we consider the Dirac–Klein–Gordon system with a mass parameter in the Dirac equation, and prove uniform (in the mass parameter) global existence, unified time decay estimates and linear scattering in the top order energy space.\u0000</p>","PeriodicalId":9478,"journal":{"name":"Calculus of Variations and Partial Differential Equations","volume":"30 1","pages":""},"PeriodicalIF":2.1,"publicationDate":"2024-08-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141937541","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}