Luigi Montoro, Luigi Muglia, Berardino Sciunzi, Domenico Vuono
{"title":"Regularity and symmetry results for the vectorial p-Laplacian","authors":"Luigi Montoro, Luigi Muglia, Berardino Sciunzi, Domenico Vuono","doi":"10.1016/j.na.2024.113700","DOIUrl":"10.1016/j.na.2024.113700","url":null,"abstract":"<div><div>We obtain some regularity results for solutions to vectorial <span><math><mi>p</mi></math></span>-Laplace equations <span><span><span><math><mrow><mo>−</mo><msub><mrow><mi>Δ</mi></mrow><mrow><mi>p</mi></mrow></msub><mi>u</mi><mo>=</mo><mo>−</mo><mi>div</mi><mrow><mo>(</mo><msup><mrow><mrow><mo>|</mo><mi>D</mi><mi>u</mi><mo>|</mo></mrow></mrow><mrow><mi>p</mi><mo>−</mo><mn>2</mn></mrow></msup><mi>D</mi><mi>u</mi><mo>)</mo></mrow><mo>=</mo><mi>f</mi><mrow><mo>(</mo><mi>x</mi><mo>,</mo><mi>u</mi><mo>)</mo></mrow><mspace></mspace><mspace></mspace><mtext>in</mtext><mi>Ω</mi><mspace></mspace><mo>.</mo></mrow></math></span></span></span>More precisely we address the issue of second order estimates for the stress field. As a consequence of our regularity results we deduce a weighted Sobolev inequality that leads to weak comparison principles. As a corollary we run over the moving plane technique to deduce symmetry and monotonicity results for the solutions, under suitable assumptions.</div></div>","PeriodicalId":49749,"journal":{"name":"Nonlinear Analysis-Theory Methods & Applications","volume":"251 ","pages":"Article 113700"},"PeriodicalIF":1.3,"publicationDate":"2024-11-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142662447","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":"Sobolev spaces for singular perturbation of 2D Laplace operator","authors":"Vladimir Georgiev , Mario Rastrelli","doi":"10.1016/j.na.2024.113710","DOIUrl":"10.1016/j.na.2024.113710","url":null,"abstract":"<div><div>We study the perturbed Sobolev space <span><math><msubsup><mrow><mi>H</mi></mrow><mrow><mi>α</mi></mrow><mrow><mn>1</mn><mo>,</mo><mi>r</mi></mrow></msubsup></math></span>, <span><math><mrow><mi>r</mi><mo>∈</mo><mrow><mo>(</mo><mn>1</mn><mo>,</mo><mi>∞</mi><mo>)</mo></mrow><mo>,</mo></mrow></math></span> associated with singular perturbation <span><math><msub><mrow><mi>Δ</mi></mrow><mrow><mi>α</mi></mrow></msub></math></span> of Laplace operator in Euclidean space of dimension <span><math><mrow><mn>2</mn><mo>.</mo></mrow></math></span> The main results give the possibility to extend the <span><math><msup><mrow><mi>L</mi></mrow><mrow><mn>2</mn></mrow></msup></math></span> theory of perturbed Sobolev space to the <span><math><msup><mrow><mi>L</mi></mrow><mrow><mi>r</mi></mrow></msup></math></span> case. When <span><math><mrow><mi>r</mi><mo>∈</mo><mrow><mo>(</mo><mn>2</mn><mo>,</mo><mi>∞</mi><mo>)</mo></mrow></mrow></math></span> we have appropriate representation of the functions in <span><math><msubsup><mrow><mi>H</mi></mrow><mrow><mi>α</mi></mrow><mrow><mn>1</mn><mo>,</mo><mi>r</mi></mrow></msubsup></math></span> in regular and singular part. An application to local well-posedness of the NLS associated with this singular perturbation in the mass critical and mass supercritical cases is established too.</div></div>","PeriodicalId":49749,"journal":{"name":"Nonlinear Analysis-Theory Methods & Applications","volume":"251 ","pages":"Article 113710"},"PeriodicalIF":1.3,"publicationDate":"2024-11-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142662446","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":"Decay characterization of weak solutions for the MHD micropolar equations on R2","authors":"Felipe W. Cruz, Mirelle M. Sousa","doi":"10.1016/j.na.2024.113701","DOIUrl":"10.1016/j.na.2024.113701","url":null,"abstract":"<div><div>We establish the characterization of decay rates of solutions to the 2D MHD micropolar system in terms of the decay character of the initial data. We also prove a faster decay rate for the micro-rotation. Moreover, we study the large time behavior of solutions by comparing them to solutions of the linear part. It is also shown that the difference between the micro-rotational field and its linear part decays faster.</div></div>","PeriodicalId":49749,"journal":{"name":"Nonlinear Analysis-Theory Methods & Applications","volume":"251 ","pages":"Article 113701"},"PeriodicalIF":1.3,"publicationDate":"2024-11-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142662448","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}
Rene Cabrera , Maria Pia Gualdani , Nestor Guillen
{"title":"Regularization estimates of the Landau–Coulomb diffusion","authors":"Rene Cabrera , Maria Pia Gualdani , Nestor Guillen","doi":"10.1016/j.na.2024.113695","DOIUrl":"10.1016/j.na.2024.113695","url":null,"abstract":"<div><div>The Landau–Coulomb equation is an important model in plasma physics featuring both nonlinear diffusion and reaction terms. In this manuscript we focus on the diffusion operator within the equation by dropping the potentially nefarious reaction term altogether. We show that the diffusion operator in the Landau–Coulomb equation provides a much stronger <span><math><mrow><msup><mrow><mi>L</mi></mrow><mrow><mn>1</mn></mrow></msup><mo>→</mo><msup><mrow><mi>L</mi></mrow><mrow><mi>∞</mi></mrow></msup></mrow></math></span> rate of regularization than its linear counterpart, the Laplace operator. The result is made possible by a nonlinear functional inequality of Gressman, Krieger, and Strain together with a De Giorgi iteration. This stronger regularization rate illustrates the importance of the nonlinear nature of the diffusion in the analysis of the Landau equation and raises the question of determining whether this rate also happens for the Landau–Coulomb equation itself.</div></div>","PeriodicalId":49749,"journal":{"name":"Nonlinear Analysis-Theory Methods & Applications","volume":"251 ","pages":"Article 113695"},"PeriodicalIF":1.3,"publicationDate":"2024-11-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142662444","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":"On the p-torsional rigidity of combinatorial graphs","authors":"Patrizio Bifulco, Delio Mugnolo","doi":"10.1016/j.na.2024.113694","DOIUrl":"10.1016/j.na.2024.113694","url":null,"abstract":"<div><div>We study the <span><math><mi>p</mi></math></span>-<em>torsion function</em> and the corresponding <span><math><mi>p</mi></math></span>-<em>torsional rigidity</em> associated with <span><math><mi>p</mi></math></span>-Laplacians and, more generally, <span><math><mi>p</mi></math></span>-Schrödinger operators, for <span><math><mrow><mn>1</mn><mo><</mo><mi>p</mi><mo><</mo><mi>∞</mi></mrow></math></span>, on possibly infinite combinatorial graphs. We present sufficient criteria for the existence of a summable <span><math><mi>p</mi></math></span>-torsion function and we derive several upper and lower bounds for the <span><math><mi>p</mi></math></span>-torsional rigidity. Our methods are mostly based on novel surgery principles. As an application, we also find some new estimates on the bottom of the spectrum of the <span><math><mi>p</mi></math></span>-Laplacian with Dirichlet conditions, thus complementing some results recently obtained in Mazón and Toledo (2023) in a more general setting. Finally, we prove a Kohler–Jobin inequality for combinatorial graphs (for <span><math><mrow><mi>p</mi><mo>=</mo><mn>2</mn></mrow></math></span>): to the best of our knowledge, graphs thus become the third ambient where a Kohler–Jobin inequality is known to hold.</div></div>","PeriodicalId":49749,"journal":{"name":"Nonlinear Analysis-Theory Methods & Applications","volume":"251 ","pages":"Article 113694"},"PeriodicalIF":1.3,"publicationDate":"2024-11-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142662493","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 existence for some fully nonlinear equations of Krylov-type arising in conformal geometry","authors":"Ya Ding, Yan He, Jun Liu","doi":"10.1016/j.na.2024.113709","DOIUrl":"10.1016/j.na.2024.113709","url":null,"abstract":"<div><div>This paper considers a class of fully nonlinear equations on Riemannian manifolds that arise in conformal geometry. Based on the a priori estimates and the blow-up analysis, we obtain the existence theorems for these equations.</div></div>","PeriodicalId":49749,"journal":{"name":"Nonlinear Analysis-Theory Methods & Applications","volume":"251 ","pages":"Article 113709"},"PeriodicalIF":1.3,"publicationDate":"2024-11-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142662495","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":"Optimal decay and regularity for a Thomas–Fermi type variational problem","authors":"Damiano Greco","doi":"10.1016/j.na.2024.113698","DOIUrl":"10.1016/j.na.2024.113698","url":null,"abstract":"<div><div>We study existence and qualitative properties of the minimizers for a Thomas–Fermi type energy functional defined by <span><span><span><math><mrow><msub><mrow><mi>E</mi></mrow><mrow><mi>α</mi></mrow></msub><mrow><mo>(</mo><mi>ρ</mi><mo>)</mo></mrow><mo>≔</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mi>q</mi></mrow></mfrac><msub><mrow><mo>∫</mo></mrow><mrow><msup><mrow><mi>R</mi></mrow><mrow><mi>d</mi></mrow></msup></mrow></msub><msup><mrow><mrow><mo>|</mo><mi>ρ</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>|</mo></mrow></mrow><mrow><mi>q</mi></mrow></msup><mi>d</mi><mi>x</mi><mo>+</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mn>2</mn></mrow></mfrac><msub><mrow><mo>∬</mo></mrow><mrow><msup><mrow><mi>R</mi></mrow><mrow><mi>d</mi></mrow></msup><mo>×</mo><msup><mrow><mi>R</mi></mrow><mrow><mi>d</mi></mrow></msup></mrow></msub><mfrac><mrow><mi>ρ</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mi>ρ</mi><mrow><mo>(</mo><mi>y</mi><mo>)</mo></mrow></mrow><mrow><msup><mrow><mrow><mo>|</mo><mi>x</mi><mo>−</mo><mi>y</mi><mo>|</mo></mrow></mrow><mrow><mi>d</mi><mo>−</mo><mi>α</mi></mrow></msup></mrow></mfrac><mi>d</mi><mi>x</mi><mi>d</mi><mi>y</mi><mo>−</mo><msub><mrow><mo>∫</mo></mrow><mrow><msup><mrow><mi>R</mi></mrow><mrow><mi>d</mi></mrow></msup></mrow></msub><mi>V</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mi>ρ</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mi>d</mi><mi>x</mi><mo>,</mo></mrow></math></span></span></span>where <span><math><mrow><mi>d</mi><mo>∈</mo><mrow><mo>[</mo><mn>2</mn><mo>,</mo><mi>∞</mi><mo>)</mo></mrow></mrow></math></span>, <span><math><mrow><mi>α</mi><mo>∈</mo><mrow><mo>(</mo><mn>0</mn><mo>,</mo><mi>d</mi><mo>)</mo></mrow></mrow></math></span>, <span><math><mrow><mi>q</mi><mo>∈</mo><mrow><mo>(</mo><mrow><mfrac><mrow><mn>2</mn><mi>d</mi></mrow><mrow><mi>d</mi><mo>+</mo><mi>α</mi></mrow></mfrac><mo>,</mo><mi>∞</mi></mrow><mo>)</mo></mrow></mrow></math></span> and <span><math><mi>V</mi></math></span> is a potential. Under broad assumptions on <span><math><mi>V</mi></math></span> we establish existence, uniqueness and qualitative properties such as positivity, regularity and decay at infinity of the global minimizer. The decay at infinity depends in a non-trivial way on the choice of <span><math><mi>α</mi></math></span> and <span><math><mi>q</mi></math></span>. If <span><math><mrow><mi>α</mi><mo>∈</mo><mrow><mo>(</mo><mn>0</mn><mo>,</mo><mn>2</mn><mo>)</mo></mrow></mrow></math></span> and <span><math><mrow><mi>q</mi><mo>></mo><mn>2</mn></mrow></math></span> the global minimizer is proved to be positive under mild regularity assumptions on <span><math><mi>V</mi></math></span>, unlike in the local case <span><math><mrow><mi>α</mi><mo>=</mo><mn>2</mn></mrow></math></span> where the global minimizer has typically compact support. We also show that if <span><math><mi>V</mi></math></span> decays sufficiently fast the global minimizer is sign-changing even if <span><math><mi>V</mi></math></span> is non-negative. In such regi","PeriodicalId":49749,"journal":{"name":"Nonlinear Analysis-Theory Methods & Applications","volume":"251 ","pages":"Article 113698"},"PeriodicalIF":1.3,"publicationDate":"2024-11-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142662449","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":"Blow-up estimates and a priori bounds for the positive solutions of a class of superlinear indefinite elliptic problems","authors":"Julián López-Gómez , Juan Carlos Sampedro","doi":"10.1016/j.na.2024.113693","DOIUrl":"10.1016/j.na.2024.113693","url":null,"abstract":"<div><div>In this paper we find out some new blow-up estimates for the positive explosive solutions of a paradigmatic class of elliptic boundary value problems of superlinear indefinite type. These estimates are obtained by combining the scaling technique of Gidas–Spruck together with a generalized De Giorgi–Moser weak Harnack inequality found, very recently, by Sirakov (2020; 2022). In a further step, based on a comparison result of Amann and López-Gómez (1998), we will show how these bounds provide us with some sharp a priori estimates for the classical positive solutions of a wide variety of superlinear indefinite problems. It turns out that this is the first general result where the decay rates of the potential in front of the nonlinearity (<span><math><mrow><mi>a</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow></mrow></math></span> in <span><span>(1.1)</span></span>) do not play any role for getting a priori bounds for the positive solutions when <span><math><mrow><mi>N</mi><mo>≥</mo><mn>3</mn></mrow></math></span>.</div></div>","PeriodicalId":49749,"journal":{"name":"Nonlinear Analysis-Theory Methods & Applications","volume":"251 ","pages":"Article 113693"},"PeriodicalIF":1.3,"publicationDate":"2024-11-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142662445","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":"p-Wasserstein barycenters","authors":"Camilla Brizzi , Gero Friesecke , Tobias Ried","doi":"10.1016/j.na.2024.113687","DOIUrl":"10.1016/j.na.2024.113687","url":null,"abstract":"<div><div>We study barycenters of <span><math><mi>N</mi></math></span> probability measures on <span><math><msup><mrow><mi>R</mi></mrow><mrow><mi>d</mi></mrow></msup></math></span> with respect to the <span><math><mi>p</mi></math></span>-Wasserstein metric (<span><math><mrow><mn>1</mn><mo><</mo><mi>p</mi><mo><</mo><mi>∞</mi></mrow></math></span>). We prove that</div><div>– <span><math><mi>p</mi></math></span>-Wasserstein barycenters of absolutely continuous measures are unique, and again absolutely continuous</div><div>– <span><math><mi>p</mi></math></span>-Wasserstein barycenters admit a multi-marginal formulation</div><div>– the optimal multi-marginal plan is unique and of Monge form if the marginals are</div><div>absolutely continuous, and its support has an explicit parametrization as a graph over any</div><div>marginal space. This extends the Agueh–Carlier theory of Wasserstein barycenters <span><span>[1]</span></span> to exponents <span><math><mrow><mi>p</mi><mo>≠</mo><mn>2</mn></mrow></math></span>. A key ingredient is a quantitative injectivity estimate for the (highly non-injective) map from <span><math><mi>N</mi></math></span>-point configurations to their <span><math><mi>p</mi></math></span>-barycenter on the support of an optimal multi-marginal plan. We also discuss the statistical meaning of <span><math><mi>p</mi></math></span>-Wasserstein barycenters in one dimension.</div></div>","PeriodicalId":49749,"journal":{"name":"Nonlinear Analysis-Theory Methods & Applications","volume":"251 ","pages":"Article 113687"},"PeriodicalIF":1.3,"publicationDate":"2024-11-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142662494","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":"Measures in the dual of BV: perimeter bounds and relations with divergence-measure fields","authors":"Giovanni E. Comi , Gian Paolo Leonardi","doi":"10.1016/j.na.2024.113686","DOIUrl":"10.1016/j.na.2024.113686","url":null,"abstract":"<div><div>We analyze some properties of the measures in the dual of the space <span><math><mrow><mi>B</mi><mi>V</mi></mrow></math></span>, by considering (signed) Radon measures satisfying a perimeter bound condition, which means that the absolute value of the measure of a set is controlled by the perimeter of the set itself, and whose total variations also belong to the dual of <span><math><mrow><mi>B</mi><mi>V</mi></mrow></math></span>. We exploit and refine the results of Cong Phuc and Torres (2017), in particular exploring the relation with divergence-measure fields and proving the stability of the perimeter bound from sets to <span><math><mrow><mi>B</mi><mi>V</mi></mrow></math></span> functions under a suitable approximation of the given measure. As an important tool, we obtain a refinement of Anzellotti-Giaquinta approximation for <span><math><mrow><mi>B</mi><mi>V</mi></mrow></math></span> functions, which is of separate interest in itself and, in the context of Anzellotti’s pairing theory for divergence-measure fields, implies a new way of approximating <span><math><mi>λ</mi></math></span>-pairings, as well as new bounds for their total variation. These results are also relevant due to their application in the study of weak solutions to the non-parametric prescribed mean curvature equation with measure data, which is explored in a subsequent work.</div></div>","PeriodicalId":49749,"journal":{"name":"Nonlinear Analysis-Theory Methods & Applications","volume":"251 ","pages":"Article 113686"},"PeriodicalIF":1.3,"publicationDate":"2024-11-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142592583","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}