Pub Date : 2024-11-16DOI: 10.1016/j.na.2024.113699
Philipp Reiser
We give new examples of manifolds that appear as cross sections of tangent cones of non-collapsed Ricci limit spaces. It was shown by Colding–Naber that the homeomorphism types of the tangent cones of a fixed point of such a space do not need to be unique. In fact, they constructed an example in dimension 5 where two different homeomorphism types appear at the same point. In this note, we extend this result and construct limit spaces in all dimensions at least 5 where any finite collection of manifolds that admit core metrics, a type of metric introduced by Perelman and Burdick to study Riemannian metrics of positive Ricci curvature on connected sums, can appear as cross sections of tangent cones of the same point.
{"title":"Examples of tangent cones of non-collapsed Ricci limit spaces","authors":"Philipp Reiser","doi":"10.1016/j.na.2024.113699","DOIUrl":"10.1016/j.na.2024.113699","url":null,"abstract":"<div><div>We give new examples of manifolds that appear as cross sections of tangent cones of non-collapsed Ricci limit spaces. It was shown by Colding–Naber that the homeomorphism types of the tangent cones of a fixed point of such a space do not need to be unique. In fact, they constructed an example in dimension 5 where two different homeomorphism types appear at the same point. In this note, we extend this result and construct limit spaces in all dimensions at least 5 where any finite collection of manifolds that admit <em>core metrics</em>, a type of metric introduced by Perelman and Burdick to study Riemannian metrics of positive Ricci curvature on connected sums, can appear as cross sections of tangent cones of the same point.</div></div>","PeriodicalId":49749,"journal":{"name":"Nonlinear Analysis-Theory Methods & Applications","volume":"252 ","pages":"Article 113699"},"PeriodicalIF":1.3,"publicationDate":"2024-11-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142656316","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}
Pub Date : 2024-11-16DOI: 10.1016/j.na.2024.113697
Piernicola Bettiol , Giuseppe De Marco , Carlo Mariconda
Consider the basic problem in the Calculus of Variations of minimizing an energy functional depending on absolutely continuous functions Under suitable assumptions on the Lagrangian, a well-known result establishes that the minimizers satisfy the Du Bois-Reymond equation. Recent work (cf. Bettiol and Mariconda, 2020 [1], 2023; Mariconda, 2023 [2], 2021, 2024) highlights not only that a Du Bois-Reymond condition for minimizers can be broadened to cover the case of nonsmooth extended valued Lagrangians, but also that a particular subdifferential (associated with the generalized Du Bois-Reymond condition) plays an important role in the approximation of the energy via its values along Lispchitz functions, no matter minimizers exist. A crucial point is establishing boundedness properties of this subdifferential, based on weak local boundedness properties of the Lagrangian. This is the main objective of this paper. Our approach is based on a refined analysis of the metric that can be employed to evaluate the distance from the complementary of the effective domain of the reference Lagrangian. As an application of our findings we show how it is possible to deduce the non-occurrence of the Lavrentiev phenomenon, providing a new general result.
考虑变分微积分的基本问题，即最小化绝对连续函数的能量函数 在拉格朗日的适当假设下，一个著名的结果确定了最小化函数满足杜布瓦-雷蒙德方程。最近的工作（参见 Bettiol 和 Mariconda，2020 [1]，2023；Mariconda，2023 [2]，2021，2024）不仅强调了最小化子的 Du Bois-Reymond 条件可以扩展到涵盖非光滑扩展值拉格朗日的情况，而且还强调了一个特定的子微分（与广义 Du Bois-Reymond 条件相关）在通过沿 Lispchitz 函数的值逼近能量方面发挥着重要作用，无论最小化子是否存在。关键的一点是根据拉格朗日的弱局部有界性特性，建立该子微分的有界性特性。这是本文的主要目标。我们的方法基于对度量的精炼分析，该度量可用于评估与参考拉格朗日有效域互补的距离。作为我们研究成果的应用，我们展示了如何推导出拉夫连季耶夫现象的不发生，从而提供了一个新的一般结果。
{"title":"A useful subdifferential in the Calculus of Variations","authors":"Piernicola Bettiol , Giuseppe De Marco , Carlo Mariconda","doi":"10.1016/j.na.2024.113697","DOIUrl":"10.1016/j.na.2024.113697","url":null,"abstract":"<div><div>Consider the basic problem in the Calculus of Variations of minimizing an energy functional depending on absolutely continuous functions Under suitable assumptions on the Lagrangian, a well-known result establishes that the minimizers satisfy the Du Bois-Reymond equation. Recent work (cf. Bettiol and Mariconda, 2020 <span><span>[1]</span></span>, 2023; Mariconda, 2023 <span><span>[2]</span></span>, 2021, 2024) highlights not only that a Du Bois-Reymond condition for minimizers can be broadened to cover the case of nonsmooth extended valued Lagrangians, but also that a particular subdifferential (associated with the generalized Du Bois-Reymond condition) plays an important role in the approximation of the energy via its values along Lispchitz functions, no matter minimizers exist. A crucial point is establishing boundedness properties of this subdifferential, based on weak local boundedness properties of the Lagrangian. This is the main objective of this paper. Our approach is based on a refined analysis of the metric that can be employed to evaluate the distance from the complementary of the effective domain of the reference Lagrangian. As an application of our findings we show how it is possible to deduce the non-occurrence of the Lavrentiev phenomenon, providing a new general result.</div></div>","PeriodicalId":49749,"journal":{"name":"Nonlinear Analysis-Theory Methods & Applications","volume":"252 ","pages":"Article 113697"},"PeriodicalIF":1.3,"publicationDate":"2024-11-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142656317","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}
Pub Date : 2024-11-15DOI: 10.1016/j.na.2024.113696
Mi-Ran Choi , Younghun Hong , Young-Ran Lee
We consider the Gabitov–Turitsyn equation or the dispersion managed nonlinear Schrödinger equation of a power-type nonlinearity and prove the global existence versus finite time blowup dichotomy for the mass-supercritical cases, that is, .
{"title":"Global existence versus finite time blowup dichotomy for the dispersion managed NLS","authors":"Mi-Ran Choi , Younghun Hong , Young-Ran Lee","doi":"10.1016/j.na.2024.113696","DOIUrl":"10.1016/j.na.2024.113696","url":null,"abstract":"<div><div>We consider the Gabitov–Turitsyn equation or the dispersion managed nonlinear Schrödinger equation of a power-type nonlinearity <span><span><span><math><mrow><mi>i</mi><msub><mrow><mi>∂</mi></mrow><mrow><mi>t</mi></mrow></msub><mi>u</mi><mo>+</mo><msub><mrow><mi>d</mi></mrow><mrow><mi>av</mi></mrow></msub><msubsup><mrow><mi>∂</mi></mrow><mrow><mi>x</mi></mrow><mrow><mn>2</mn></mrow></msubsup><mi>u</mi><mo>+</mo><msubsup><mrow><mo>∫</mo></mrow><mrow><mn>0</mn></mrow><mrow><mn>1</mn></mrow></msubsup><msup><mrow><mi>e</mi></mrow><mrow><mo>−</mo><mi>i</mi><mi>r</mi><msubsup><mrow><mi>∂</mi></mrow><mrow><mi>x</mi></mrow><mrow><mn>2</mn></mrow></msubsup></mrow></msup><mrow><mo>(</mo><mrow><msup><mrow><mrow><mo>|</mo><msup><mrow><mi>e</mi></mrow><mrow><mi>i</mi><mi>r</mi><msubsup><mrow><mi>∂</mi></mrow><mrow><mi>x</mi></mrow><mrow><mn>2</mn></mrow></msubsup></mrow></msup><mi>u</mi><mo>|</mo></mrow></mrow><mrow><mi>p</mi><mo>−</mo><mn>1</mn></mrow></msup><msup><mrow><mi>e</mi></mrow><mrow><mi>i</mi><mi>r</mi><msubsup><mrow><mi>∂</mi></mrow><mrow><mi>x</mi></mrow><mrow><mn>2</mn></mrow></msubsup></mrow></msup><mi>u</mi></mrow><mo>)</mo></mrow><mi>d</mi><mi>r</mi><mo>=</mo><mn>0</mn></mrow></math></span></span></span>and prove the global existence versus finite time blowup dichotomy for the mass-supercritical cases, that is, <span><math><mrow><mi>p</mi><mo>></mo><mn>9</mn></mrow></math></span>.</div></div>","PeriodicalId":49749,"journal":{"name":"Nonlinear Analysis-Theory Methods & Applications","volume":"251 ","pages":"Article 113696"},"PeriodicalIF":1.3,"publicationDate":"2024-11-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142662496","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 : 2024-11-15DOI: 10.1016/j.na.2024.113700
Luigi Montoro, Luigi Muglia, Berardino Sciunzi, Domenico Vuono
We obtain some regularity results for solutions to vectorial -Laplace equations 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.
{"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}
Pub Date : 2024-11-15DOI: 10.1016/j.na.2024.113710
Vladimir Georgiev , Mario Rastrelli
We study the perturbed Sobolev space , associated with singular perturbation of Laplace operator in Euclidean space of dimension The main results give the possibility to extend the theory of perturbed Sobolev space to the case. When we have appropriate representation of the functions in 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.
{"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}
Pub Date : 2024-11-15DOI: 10.1016/j.na.2024.113701
Felipe W. Cruz, Mirelle M. Sousa
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.
{"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}
Pub Date : 2024-11-13DOI: 10.1016/j.na.2024.113695
Rene Cabrera , Maria Pia Gualdani , Nestor Guillen
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 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.
朗道-库仑方程是等离子体物理学中的一个重要模型，同时具有非线性扩散和反应项。在本手稿中，我们放弃了潜在的有害反应项，将重点放在方程中的扩散算子上。我们的研究表明，朗道-库仑方程中的扩散算子比其线性对应的拉普拉斯算子具有更强的 L1→L∞ 正则化率。这一结果得益于 Gressman、Krieger 和 Strain 的非线性函数不等式以及 De Giorgi 迭代。这种更强的正则化率说明了扩散的非线性性质在朗道方程分析中的重要性，并提出了确定朗道-库仑方程本身是否也会出现这种正则化率的问题。
{"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}
Pub Date : 2024-11-12DOI: 10.1016/j.na.2024.113694
Patrizio Bifulco, Delio Mugnolo
We study the -torsion function and the corresponding -torsional rigidity associated with -Laplacians and, more generally, -Schrödinger operators, for , on possibly infinite combinatorial graphs. We present sufficient criteria for the existence of a summable -torsion function and we derive several upper and lower bounds for the -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 -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 ): to the best of our knowledge, graphs thus become the third ambient where a Kohler–Jobin inequality is known to hold.
{"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}
Pub Date : 2024-11-12DOI: 10.1016/j.na.2024.113709
Ya Ding, Yan He, Jun Liu
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.
{"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}
Pub Date : 2024-11-12DOI: 10.1016/j.na.2024.113698
Damiano Greco
<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
我们研究由 Eα(ρ)≔1q∫Rd|ρ(x)|qdx+12∬Rd×Rdρ(x)ρ(y)|x-y|d-αdxdy-∫RdV(x)ρ(x)dx 定义的托马斯-费米型能量函数的最小值的存在性和定性性质、其中 d∈[2,∞)，α∈(0,d)，q∈(2dd+α,∞)，V 是一个势。根据对 V 的宽泛假设，我们确定了全局最小值的存在性、唯一性和定性特性，如正向性、正则性和无穷大时的衰减。如果α∈(0,2)和q>2，全局最小值在 V 的温和正则性假设下被证明为正值，这与局部情况 α=2 不同，在局部情况下，全局最小值具有典型的紧凑支持。我们还证明，如果 V 的衰减速度足够快，即使 V 为非负，全局最小值也会发生符号变化。在这种情况下，我们建立了全局最小值的正向部分与约束于非负函数锥的能量最小值的支持之间的关系。我们的研究受到石墨烯中电荷筛选的最新模型的启发，在这些模型中，符号变化最小化以一种自然的方式出现。
{"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}
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