Archive for Rational Mechanics and Analysis最新文献

{"title":"Critical Perturbations for Second Order Elliptic Operators—Part II: Non-tangential Maximal Function Estimates","authors":"Simon Bortz,&nbsp;Steve Hofmann,&nbsp;José Luis Luna Garcia,&nbsp;Svitlana Mayboroda,&nbsp;Bruno Poggi","doi":"10.1007/s00205-024-01977-x","DOIUrl":"10.1007/s00205-024-01977-x","url":null,"abstract":"<div><p>This is the final part of a series of papers where we study perturbations of divergence form second order elliptic operators <span>(-textrm{div}A nabla )</span> by first and zero order terms, whose complex coefficients lie in critical spaces, via the method of layer potentials. In particular, we show that the <span>(L^2)</span> well-posedness (with natural non-tangential maximal function estimates) of the Dirichlet, Neumann and regularity problems for complex Hermitian, block form, or constant-coefficient divergence form elliptic operators in the upper half-space are all stable under such perturbations. Due to the lack of the classical De Giorgi–Nash–Moser theory in our setting, our method to prove the non-tangential maximal function estimates relies on a completely new argument: We obtain a certain weak-<span>(L^p)</span> “<span>(N&lt;S)</span>” estimate, which we eventually couple with square function bounds, weighted extrapolation theory, and a bootstrapping argument to recover the full <span>(L^2)</span> bound. Finally, we show the existence and uniqueness of solutions in a relatively broad class. As a corollary, we claim the first results in an unbounded domain concerning the <span>(L^p)</span>-solvability of boundary value problems for the magnetic Schrödinger operator <span>(-(nabla -itextbf{a})^2+V)</span> when the magnetic potential <span>(textbf{a})</span> and the electric potential <i>V</i> are accordingly small in the norm of a scale-invariant Lebesgue space.\u0000</p></div>","PeriodicalId":55484,"journal":{"name":"Archive for Rational Mechanics and Analysis","volume":null,"pages":null},"PeriodicalIF":2.6,"publicationDate":"2024-04-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140564849","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Existence of Homogeneous Euler Flows of Degree (-alpha notin [-2,0])","authors":"Ken Abe","doi":"10.1007/s00205-024-01974-0","DOIUrl":"10.1007/s00205-024-01974-0","url":null,"abstract":"&lt;div&gt;&lt;p&gt;We consider (&lt;span&gt;(-alpha )&lt;/span&gt;)-homogeneous solutions to the stationary incompressible Euler equations in &lt;span&gt;({mathbb {R}}^{3}backslash {0})&lt;/span&gt; for &lt;span&gt;(alpha geqq 0)&lt;/span&gt; and in &lt;span&gt;({mathbb {R}}^{3})&lt;/span&gt; for &lt;span&gt;(alpha &lt;0)&lt;/span&gt;. Shvydkoy (2018) demonstrated the &lt;i&gt;nonexistence&lt;/i&gt; of (&lt;span&gt;(-1)&lt;/span&gt;)-homogeneous solutions &lt;span&gt;((u,p)in C^{1}({mathbb {R}}^{3}backslash {0}))&lt;/span&gt; and (&lt;span&gt;(-alpha )&lt;/span&gt;)-homogeneous solutions in the range &lt;span&gt;(0leqq alpha leqq 2)&lt;/span&gt; for the Beltrami and axisymmetric flow; namely, that no (&lt;span&gt;(-alpha )&lt;/span&gt;)-homogeneous solutions &lt;span&gt;((u,p)in C^{1}({mathbb {R}}^{3}backslash {0}))&lt;/span&gt; for &lt;span&gt;(1leqq alpha leqq 2)&lt;/span&gt; and &lt;span&gt;((u,p)in C^{2}({mathbb {R}}^{3}backslash {0}))&lt;/span&gt; for &lt;span&gt;(0leqq alpha &lt; 1)&lt;/span&gt; exist among these particular classes of flows other than irrotational solutions for integers &lt;span&gt;(alpha )&lt;/span&gt;. The nonexistence result of the Beltrami (&lt;span&gt;(-alpha )&lt;/span&gt;)-homogeneous solutions &lt;span&gt;((u,p)in C^{2}({mathbb {R}}^{3}backslash {0}))&lt;/span&gt; holds for all &lt;span&gt;(alpha &lt;1)&lt;/span&gt;. We show the nonexistence of axisymmetric (&lt;span&gt;(-alpha )&lt;/span&gt;)-homogeneous solutions without swirls &lt;span&gt;((u,p)in C^{2}({mathbb {R}}^{3}backslash {0}))&lt;/span&gt; for &lt;span&gt;(-2leqq alpha &lt;0)&lt;/span&gt;. The main result of this study is the &lt;i&gt;existence&lt;/i&gt; of axisymmetric (&lt;span&gt;(-alpha )&lt;/span&gt;)-homogeneous solutions in the complementary range &lt;span&gt;(alpha in {mathbb {R}}backslash [0,2])&lt;/span&gt;. More specifically, we show the existence of axisymmetric Beltrami (&lt;span&gt;(-alpha )&lt;/span&gt;)-homogeneous solutions &lt;span&gt;((u,p)in C^{1}({mathbb {R}}^{3}backslash {0}))&lt;/span&gt; for &lt;span&gt;(alpha &gt;2)&lt;/span&gt; and &lt;span&gt;((u,p)in C({mathbb {R}}^{3}))&lt;/span&gt; for &lt;span&gt;(alpha &lt;0)&lt;/span&gt; and axisymmetric (&lt;span&gt;(-alpha )&lt;/span&gt;)-homogeneous solutions with a nonconstant Bernoulli function &lt;span&gt;((u,p)in C^{1}({mathbb {R}}^{3}backslash {0}))&lt;/span&gt; for &lt;span&gt;(alpha &gt;2)&lt;/span&gt; and &lt;span&gt;((u,p)in C({mathbb {R}}^{3}))&lt;/span&gt; for &lt;span&gt;(alpha &lt;-2)&lt;/span&gt;, including axisymmetric (&lt;span&gt;(-alpha )&lt;/span&gt;)-homogeneous solutions without swirls &lt;span&gt;((u,p)in C^{2}({mathbb {R}}^{3}backslash {0}))&lt;/span&gt; for &lt;span&gt;(alpha &gt;2)&lt;/span&gt; and &lt;span&gt;((u,p)in C^{1}({mathbb {R}}^{3}backslash {0})cap C({mathbb {R}}^{3}))&lt;/span&gt; for &lt;span&gt;(alpha &lt;-2)&lt;/span&gt;. This is the first existence result on (&lt;span&gt;(-alpha )&lt;/span&gt;)-homogeneous solutions with no explicit forms. The level sets of the axisymmetric stream function of the irrotational (&lt;span&gt;(-alpha )&lt;/span&gt;)-homogeneous solutions in the cross-section are the Jordan curves for &lt;span&gt;(alpha =3)&lt;/span&gt;. For &lt;span&gt;(2&lt;alpha &lt;3)&lt;/span&gt;, we show the existence of axisymmetric (&lt;span&gt;(-alpha )&lt;/span&gt;)-homogeneous solutions whose stream function level sets are the Jordan curves. They provide new examples of the Beltrami/Euler flows in &lt;span&gt;({mathbb {R}}^{3}backslash {0})&lt;/span&gt; whose level sets of the proportionality ","PeriodicalId":55484,"journal":{"name":"Archive for Rational Mechanics and Analysis","volume":null,"pages":null},"PeriodicalIF":2.6,"publicationDate":"2024-04-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140564931","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Higher Order Boundary Harnack Principle via Degenerate Equations","authors":"Susanna Terracini,&nbsp;Giorgio Tortone,&nbsp;Stefano Vita","doi":"10.1007/s00205-024-01973-1","DOIUrl":"10.1007/s00205-024-01973-1","url":null,"abstract":"<div><p>As a first result we prove higher order Schauder estimates for solutions to singular/degenerate elliptic equations of type </p><div><div><span>$$begin{aligned} -textrm{div}left( rho ^aAnabla wright) =rho ^af+textrm{div}left( rho ^aFright) quad text {in}; Omega end{aligned}$$</span></div></div><p>for exponents <span>(a&gt;-1)</span>, where the weight <span>(rho )</span> vanishes with non zero gradient on a regular hypersurface <span>(Gamma )</span>, which can be either a part of the boundary of <span>(Omega )</span> or mostly contained in its interior. As an application, we extend such estimates to the ratio <i>v</i>/<i>u</i> of two solutions to a second order elliptic equation in divergence form when the zero set of <i>v</i> includes the zero set of <i>u</i> which is not singular in the domain (in this case <span>(rho =u)</span>, <span>(a=2)</span> and <span>(w=v/u)</span>). We prove first the <span>(C^{k,alpha })</span>-regularity of the ratio from one side of the regular part of the nodal set of <i>u</i> in the spirit of the higher order boundary Harnack principle in Savin (Discrete Contin Dyn Syst 35–12:6155–6163, 2015). Then, by a gluing Lemma, the estimates extend across the regular part of the nodal set. Finally, using conformal mapping in dimension <span>(n=2)</span>, we provide local gradient estimates for the ratio, which hold also across the singular set.</p></div>","PeriodicalId":55484,"journal":{"name":"Archive for Rational Mechanics and Analysis","volume":null,"pages":null},"PeriodicalIF":2.6,"publicationDate":"2024-03-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140564854","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Towards the Optimality of the Ball for the Rayleigh Conjecture Concerning the Clamped Plate","authors":"Roméo Leylekian","doi":"10.1007/s00205-024-01972-2","DOIUrl":"10.1007/s00205-024-01972-2","url":null,"abstract":"<div><p>The first eigenvalue of the Dirichlet bilaplacian shall be interpreted as the principal frequency of a vibrating plate with clamped boundary. In 1894, Rayleigh conjectured that, upon prescribing the area, the vibrating clamped plate with least principal frequency is circular. In 1995, Nadirashvili proved the Rayleigh Conjecture. Subsequently, Ashbaugh and Benguria proved the analogue of the conjecture in dimension 3. Since then, the conjecture has remained open in dimension <span>(d&gt;3)</span>. In this document, we contribute in answering the conjecture in high dimension under a particular assumption regarding the critical values of the optimal eigenfunction. More precisely, we prove that if the optimal eigenfunction has no critical value except its minimum and maximum, then the conjecture holds. This is performed thanks to an improvement of Talenti’s comparison principle, made possible after a fine study of the geometry of the eigenfunction’s nodal domains.</p></div>","PeriodicalId":55484,"journal":{"name":"Archive for Rational Mechanics and Analysis","volume":null,"pages":null},"PeriodicalIF":2.6,"publicationDate":"2024-03-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140313234","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"On the Nodal Set of Solutions to Some Sublinear Equations Without Homogeneity","authors":"Nicola Soave,&nbsp;Giorgio Tortone","doi":"10.1007/s00205-024-01970-4","DOIUrl":"10.1007/s00205-024-01970-4","url":null,"abstract":"<div><p>We investigate the structure of the nodal set of solutions to an unstable Alt-Philips type problem </p><div><div><span>$$begin{aligned} -Delta u = lambda _+(u^+)^{p-1}-lambda _-(u^-)^{q-1}, end{aligned}$$</span></div></div><p>where <span>(1 le p&lt;q&lt;2)</span>, <span>(lambda _+ &gt;0)</span>, <span>(lambda _- ge 0)</span>. The equation is characterized by the sublinear <i>inhomogeneous</i> character of the right hand-side, which makes it difficult to adapt in a standard way classical tools from free-boundary problems, such as monotonicity formulas and blow-up arguments. Our main results are: the local behavior of solutions close to the nodal set; the complete classification of the admissible vanishing orders, and estimates on the Hausdorff dimension of the singular set, for local minimizers; the existence of degenerate (not locally minimal) solutions.</p></div>","PeriodicalId":55484,"journal":{"name":"Archive for Rational Mechanics and Analysis","volume":null,"pages":null},"PeriodicalIF":2.6,"publicationDate":"2024-03-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140302053","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Invariant Manifolds for the Thin Film Equation","authors":"Christian Seis,&nbsp;Dominik Winkler","doi":"10.1007/s00205-024-01968-y","DOIUrl":"10.1007/s00205-024-01968-y","url":null,"abstract":"<div><p>The large-time behavior of solutions to the thin film equation with linear mobility in the complete wetting regime on <span>(mathbb {R}^N)</span> is examined. We investigate the higher order asymptotics of solutions converging towards self-similar Smyth–Hill solutions under certain symmetry assumptions on the initial data. The analysis is based on a construction of finite-dimensional invariant manifolds that solutions approximate to an arbitrarily prescribed order.</p></div>","PeriodicalId":55484,"journal":{"name":"Archive for Rational Mechanics and Analysis","volume":null,"pages":null},"PeriodicalIF":2.6,"publicationDate":"2024-03-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s00205-024-01968-y.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140302171","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Perturbation at Blow-Up Time of Self-Similar Solutions for the Modified Korteweg–de Vries Equation","authors":"Simão Correia,&nbsp;Raphaël Côte","doi":"10.1007/s00205-024-01969-x","DOIUrl":"10.1007/s00205-024-01969-x","url":null,"abstract":"<div><p>We prove a first stability result of self-similar blow-up for the modified Korteweg–de Vries equation on the line. More precisely, given a self-similar solution and a sufficiently small regular profile, there is a unique global solution which behaves at <span>(t=0)</span> as the sum of the self-similar solution and the smooth perturbation.</p></div>","PeriodicalId":55484,"journal":{"name":"Archive for Rational Mechanics and Analysis","volume":null,"pages":null},"PeriodicalIF":2.6,"publicationDate":"2024-03-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140199433","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"On the Triple Junction Problem without Symmetry Hypotheses","authors":"Nicholas D. Alikakos,&nbsp;Zhiyuan Geng","doi":"10.1007/s00205-024-01966-0","DOIUrl":"10.1007/s00205-024-01966-0","url":null,"abstract":"<div><p>We investigate the Allen–Cahn system <span>(Delta u-W_u(u)=0)</span>, <span>(u:mathbb {R}^2rightarrow mathbb {R}^2)</span>, where <span>(Win C^2(mathbb {R}^2,[0,+infty )))</span> is a potential with three global minima. We establish the existence of an entire solution <i>u</i> which possesses a triple junction structure. The main strategy is to study the global minimizer <span>(u_varepsilon )</span> of the variational problem <span>(min int _{B_1} left( frac{varepsilon }{2}vert nabla uvert ^2+frac{1}{varepsilon }W(u) right) ,textrm{d}z)</span>, <span>(u=g_varepsilon )</span> on <span>(partial B_1)</span> for some suitable boundary data <span>(g_varepsilon )</span>. The point of departure is an energy lower bound that plays a crucial role in estimating the location and size of the diffuse interface. We do not impose any symmetry hypothesis on the solution or on the potential.</p></div>","PeriodicalId":55484,"journal":{"name":"Archive for Rational Mechanics and Analysis","volume":null,"pages":null},"PeriodicalIF":2.6,"publicationDate":"2024-03-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140199489","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Self-Similar Finite-Time Blowups with Smooth Profiles of the Generalized Constantin–Lax–Majda Model","authors":"De Huang,&nbsp;Xiang Qin,&nbsp;Xiuyuan Wang,&nbsp;Dongyi Wei","doi":"10.1007/s00205-024-01971-3","DOIUrl":"10.1007/s00205-024-01971-3","url":null,"abstract":"<div><p>We show that the <i>a</i>-parameterized family of the generalized Constantin–Lax–Majda model, also known as the Okamoto–Sakajo–Wunsch model, admits exact self-similar finite-time blowup solutions with interiorly smooth profiles for all <span>(ale 1)</span>. Depending on the value of <i>a</i>, these self-similar profiles are either smooth on the whole real line or compactly supported and smooth in the interior of their closed supports. The existence of these profiles is proved in a consistent way by considering the fixed-point problem of an <i>a</i>-dependent nonlinear map, based on which detailed characterizations of their regularity, monotonicity, and far-field decay rates are established. Our work unifies existing results for some discrete values of <i>a</i> and also explains previous numerical observations for a wide range of <i>a</i>.\u0000</p></div>","PeriodicalId":55484,"journal":{"name":"Archive for Rational Mechanics and Analysis","volume":null,"pages":null},"PeriodicalIF":2.6,"publicationDate":"2024-03-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140199325","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Higher Order Goh Conditions for Singular Extremals of Corank 1","authors":"Francesco Boarotto,&nbsp;Roberto Monti,&nbsp;Alessandro Socionovo","doi":"10.1007/s00205-024-01964-2","DOIUrl":"10.1007/s00205-024-01964-2","url":null,"abstract":"<div><p>We prove Goh conditions of order <span>(ngeqq 3)</span> for strictly singular length-minimizing curves of corank 1, under the assumption that the domain of the <i>n</i>th instrinsic differential is of finite codimension. This result relies upon the proof of an open mapping theorem for maps with a regular <i>n</i>th differential.</p></div>","PeriodicalId":55484,"journal":{"name":"Archive for Rational Mechanics and Analysis","volume":null,"pages":null},"PeriodicalIF":2.6,"publicationDate":"2024-03-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140301989","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}