Pub Date : 2025-07-21DOI: 10.1016/j.matpur.2025.103778
Fabio Nicola
We focus on quantum systems represented by a Hilbert space , where A is a locally compact Abelian group that contains a compact open subgroup. We examine two interconnected issues related to Weyl-Heisenberg operators. First, we provide a complete and elegant solution to the problem of describing the stabilizer states in terms of their wave functions — an issue that arises in quantum information theory. Subsequently, we demonstrate that the stabilizer states are exactly the minimizers of the Wehrl entropy, thereby solving the Wehrl-type entropy conjecture for any such group (in particular, for finite-dimensional vector spaces over non-Archimedean local fields). Additionally, we construct a moduli space for the set of stabilizer states, that is, a parametrization of this set, that endows it with a natural algebraic structure, and we derive a formula for the number of stabilizer states when A is finite. Indeed, these results are novel even for finite Abelian groups.
{"title":"The wave function of stabilizer states and the Wehrl conjecture","authors":"Fabio Nicola","doi":"10.1016/j.matpur.2025.103778","DOIUrl":"10.1016/j.matpur.2025.103778","url":null,"abstract":"<div><div>We focus on quantum systems represented by a Hilbert space <span><math><msup><mrow><mi>L</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>(</mo><mi>A</mi><mo>)</mo></math></span>, where <em>A</em> is a locally compact Abelian group that contains a compact open subgroup. We examine two interconnected issues related to Weyl-Heisenberg operators. First, we provide a complete and elegant solution to the problem of describing the stabilizer states in terms of their wave functions — an issue that arises in quantum information theory. Subsequently, we demonstrate that the stabilizer states are exactly the minimizers of the Wehrl entropy, thereby solving the Wehrl-type entropy conjecture for any such group (in particular, for finite-dimensional vector spaces over non-Archimedean local fields). Additionally, we construct a moduli space for the set of stabilizer states, that is, a parametrization of this set, that endows it with a natural algebraic structure, and we derive a formula for the number of stabilizer states when <em>A</em> is finite. Indeed, these results are novel even for finite Abelian groups.</div></div>","PeriodicalId":51071,"journal":{"name":"Journal de Mathematiques Pures et Appliquees","volume":"205 ","pages":"Article 103778"},"PeriodicalIF":2.3,"publicationDate":"2025-07-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144748574","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}
Pub Date : 2025-07-21DOI: 10.1016/j.matpur.2025.103771
Alex Abreu , Antonio Nigro
The characters of Kazhdan–Lusztig elements of the Hecke algebra over (and in particular, the chromatic symmetric function of indifference graphs) are completely encoded in the (intersection) cohomology of Lusztig varieties. Considering the forgetful map to some partial flag variety, the decomposition theorem tells us that this cohomology splits as a sum of intersection cohomology groups with coefficients in some local systems of subvarieties of the partial flag variety. We prove that these local systems correspond to representations of subgroups of . An explicit characterization of such representations would provide a recursive formula for the computation of such characters/chromatic symmetric functions, which could settle Haiman's conjecture about the positivity of the monomial characters of Kazhdan–Lusztig elements and Stanley–Stembridge conjecture about e-positivity of chromatic symmetric function of indifference graphs. We also find a connection between the character of certain homology groups of subvarieties of the partial flag varieties and the Grojnowski–Haiman hybrid basis of the Hecke algebra.
{"title":"Parabolic Lusztig varieties and chromatic symmetric functions","authors":"Alex Abreu , Antonio Nigro","doi":"10.1016/j.matpur.2025.103771","DOIUrl":"10.1016/j.matpur.2025.103771","url":null,"abstract":"<div><div>The characters of Kazhdan–Lusztig elements of the Hecke algebra over <span><math><msub><mrow><mi>S</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span> (and in particular, the chromatic symmetric function of indifference graphs) are completely encoded in the (intersection) cohomology of Lusztig varieties. Considering the forgetful map to some partial flag variety, the decomposition theorem tells us that this cohomology splits as a sum of intersection cohomology groups with coefficients in some local systems of subvarieties of the partial flag variety. We prove that these local systems correspond to representations of subgroups of <span><math><msub><mrow><mi>S</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span>. An explicit characterization of such representations would provide a recursive formula for the computation of such characters/chromatic symmetric functions, which could settle Haiman's conjecture about the positivity of the monomial characters of Kazhdan–Lusztig elements and Stanley–Stembridge conjecture about <em>e</em>-positivity of chromatic symmetric function of indifference graphs. We also find a connection between the character of certain homology groups of subvarieties of the partial flag varieties and the Grojnowski–Haiman hybrid basis of the Hecke algebra.</div></div>","PeriodicalId":51071,"journal":{"name":"Journal de Mathematiques Pures et Appliquees","volume":"203 ","pages":"Article 103771"},"PeriodicalIF":2.1,"publicationDate":"2025-07-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144696692","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}
Pub Date : 2025-07-21DOI: 10.1016/j.matpur.2025.103775
Zhi Hu , Pengfei Huang , Runhong Zong
This paper considers the moduli spaces/stacks of parabolic bundles (parabolic logarithmic flat bundles and parabolic logarithmic Higgs bundles with given spectrum) of rank 2 and degree 1 over with five marked points. The foliation and stratification structures on these moduli spaces/stacks are investigated. In particular, we confirm Simpson's conjecture for the moduli space of parabolic logarithmic flat bundles with certain non-special weight system.
{"title":"Moduli spaces of parabolic bundles over P1 with five marked points","authors":"Zhi Hu , Pengfei Huang , Runhong Zong","doi":"10.1016/j.matpur.2025.103775","DOIUrl":"10.1016/j.matpur.2025.103775","url":null,"abstract":"<div><div>This paper considers the moduli spaces/stacks of parabolic bundles (parabolic logarithmic flat bundles and parabolic logarithmic Higgs bundles with given spectrum) of rank 2 and degree 1 over <span><math><msup><mrow><mi>P</mi></mrow><mrow><mn>1</mn></mrow></msup></math></span> with five marked points. The foliation and stratification structures on these moduli spaces/stacks are investigated. In particular, we confirm Simpson's conjecture for the moduli space of parabolic logarithmic flat bundles with certain non-special weight system.</div></div>","PeriodicalId":51071,"journal":{"name":"Journal de Mathematiques Pures et Appliquees","volume":"205 ","pages":"Article 103775"},"PeriodicalIF":2.3,"publicationDate":"2025-07-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144739251","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}
Pub Date : 2025-07-18DOI: 10.1016/j.matpur.2025.103774
Giuseppe Maria Coclite , Nicola De Nitti , Mauro Garavello , Francesca Marcellini
We consider the p-system in Eulerian coordinates on a star-shaped network. Under suitable transmission conditions at the junction and dissipative boundary conditions at the exterior vertices, we show that the entropy solutions of the system are exponentially stabilizable. Our proof extends the strategy by Coron et al. (2017) and is based on a front-tracking algorithm used to construct approximate piecewise constant solutions whose BV norms are controlled through a suitable exponentially-weighted Glimm-type Lyapunov functional.
{"title":"Feedback stabilization for entropy solutions of a 2 × 2 hyperbolic system of conservation laws at a junction","authors":"Giuseppe Maria Coclite , Nicola De Nitti , Mauro Garavello , Francesca Marcellini","doi":"10.1016/j.matpur.2025.103774","DOIUrl":"10.1016/j.matpur.2025.103774","url":null,"abstract":"<div><div>We consider the <em>p</em>-system in Eulerian coordinates on a star-shaped network. Under suitable transmission conditions at the junction and dissipative boundary conditions at the exterior vertices, we show that the entropy solutions of the system are exponentially stabilizable. Our proof extends the strategy by Coron et al. (2017) and is based on a front-tracking algorithm used to construct approximate piecewise constant solutions whose BV norms are controlled through a suitable exponentially-weighted Glimm-type Lyapunov functional.</div></div>","PeriodicalId":51071,"journal":{"name":"Journal de Mathematiques Pures et Appliquees","volume":"205 ","pages":"Article 103774"},"PeriodicalIF":2.3,"publicationDate":"2025-07-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144779585","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}
Pub Date : 2025-06-16DOI: 10.1016/j.matpur.2025.103750
Frédéric Charve
The asymptotics of the strongly stratified Boussinesq system when the Froude number goes to zero have been previously investigated, but the resulting limit system surprisingly did not depend on the thermal diffusivity . In this article we obtain richer asymptotics (depending on ) for more general ill-prepared initial data.
As for the rotating fluids system, the only way to reach this limit consists in finding suitable non-conventional initial data: here, to a function classically depending on the full space variable, we add a second one only depending on the vertical coordinate.
Thanks to a refined study of the structure of the limit system and to new adapted Strichartz estimates, we obtain convergence in the context of weak Leray-type solutions providing explicit convergence rates when possible. In the usually simpler case we are able to improve the Strichartz estimates and the convergence rates. The last part of the appendix is devoted to the proof of a new and crucial dispersion estimate, as classical methods fail.
Finally, our theorems can also be rewritten as a global existence result and asymptotic expansion for the classical Boussinesq system near an explicit stationary solution and for large non-conventional vertically stratified initial data.
{"title":"Hidden asymptotics for the weak solutions of the strongly stratified Boussinesq system without rotation","authors":"Frédéric Charve","doi":"10.1016/j.matpur.2025.103750","DOIUrl":"10.1016/j.matpur.2025.103750","url":null,"abstract":"<div><div>The asymptotics of the strongly stratified Boussinesq system when the Froude number goes to zero have been previously investigated, but the resulting limit system surprisingly did not depend on the thermal diffusivity <span><math><msup><mrow><mi>ν</mi></mrow><mrow><mo>′</mo></mrow></msup></math></span>. In this article we obtain richer asymptotics (depending on <span><math><msup><mrow><mi>ν</mi></mrow><mrow><mo>′</mo></mrow></msup></math></span>) for more general ill-prepared initial data.</div><div>As for the rotating fluids system, the only way to reach this limit consists in finding suitable non-conventional initial data: here, to a function classically depending on the full space variable, we add a second one only depending on the vertical coordinate.</div><div>Thanks to a refined study of the structure of the limit system and to new adapted Strichartz estimates, we obtain convergence in the context of weak Leray-type solutions providing explicit convergence rates when possible. In the usually simpler case <span><math><mi>ν</mi><mo>=</mo><msup><mrow><mi>ν</mi></mrow><mrow><mo>′</mo></mrow></msup></math></span> we are able to improve the Strichartz estimates and the convergence rates. The last part of the appendix is devoted to the proof of a new and crucial dispersion estimate, as classical methods fail.</div><div>Finally, our theorems can also be rewritten as a global existence result and asymptotic expansion for the classical Boussinesq system near an explicit stationary solution and for large non-conventional vertically stratified initial data.</div></div>","PeriodicalId":51071,"journal":{"name":"Journal de Mathematiques Pures et Appliquees","volume":"202 ","pages":"Article 103750"},"PeriodicalIF":2.1,"publicationDate":"2025-06-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144313661","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}
Pub Date : 2025-06-16DOI: 10.1016/j.matpur.2025.103751
Rufat Badal , Manuel Friedrich , Martin Kružík , Lennart Machill
According to the Nernst theorem or, equivalently, the third law of thermodynamics, the absolute zero temperature is not attainable. Starting with an initial positive temperature, we show that there exist solutions to a Kelvin-Voigt model for quasi-static nonlinear thermoviscoelasticity at a finite-strain setting [45], obeying an exponential-in-time lower bound on the temperature. Afterwards, we focus on the case of deformations near the identity and temperatures near a critical positive temperature, and we show that weak solutions of the nonlinear system converge in a suitable sense to solutions of a system in linearized thermoviscoelasticity. Our result extends the recent linearization result in [4], as it allows the critical temperature to be positive.
{"title":"Positive temperature in nonlinear thermoviscoelasticity and the derivation of linearized models","authors":"Rufat Badal , Manuel Friedrich , Martin Kružík , Lennart Machill","doi":"10.1016/j.matpur.2025.103751","DOIUrl":"10.1016/j.matpur.2025.103751","url":null,"abstract":"<div><div>According to the Nernst theorem or, equivalently, the third law of thermodynamics, the absolute zero temperature is not attainable. Starting with an initial positive temperature, we show that there exist solutions to a Kelvin-Voigt model for quasi-static nonlinear thermoviscoelasticity at a finite-strain setting <span><span>[45]</span></span>, obeying an exponential-in-time lower bound on the temperature. Afterwards, we focus on the case of deformations near the identity and temperatures near a critical positive temperature, and we show that weak solutions of the nonlinear system converge in a suitable sense to solutions of a system in linearized thermoviscoelasticity. Our result extends the recent linearization result in <span><span>[4]</span></span>, as it allows the critical temperature to be positive.</div></div>","PeriodicalId":51071,"journal":{"name":"Journal de Mathematiques Pures et Appliquees","volume":"202 ","pages":"Article 103751"},"PeriodicalIF":2.1,"publicationDate":"2025-06-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144322454","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}
Pub Date : 2025-06-16DOI: 10.1016/j.matpur.2025.103752
Jiajun Tong , Yuming Paul Zhang
We investigate the general Porous Medium Equations with drift and source terms that model tumor growth. Incompressible limit of such models has been well-studied in the literature, where convergence of the density and pressure variables are established, while it remains unclear whether the free boundaries of the solutions exhibit convergence as well. In this paper, we provide an affirmative result by showing that the free boundaries converge in the Hausdorff distance in the incompressible limit. To achieve this, we quantify the relation between the free boundary motion and spatial average of the pressure, and establish a uniform-in-m strict expansion property of the pressure supports. As a corollary, we derive upper bounds for the Hausdorff dimensions of the free boundaries and show that the limiting free boundary has finite -dimensional Hausdorff measure.
{"title":"Convergence of free boundaries in the incompressible limit of tumor growth models","authors":"Jiajun Tong , Yuming Paul Zhang","doi":"10.1016/j.matpur.2025.103752","DOIUrl":"10.1016/j.matpur.2025.103752","url":null,"abstract":"<div><div>We investigate the general Porous Medium Equations with drift and source terms that model tumor growth. Incompressible limit of such models has been well-studied in the literature, where convergence of the density and pressure variables are established, while it remains unclear whether the free boundaries of the solutions exhibit convergence as well. In this paper, we provide an affirmative result by showing that the free boundaries converge in the Hausdorff distance in the incompressible limit. To achieve this, we quantify the relation between the free boundary motion and spatial average of the pressure, and establish a uniform-in-<em>m</em> strict expansion property of the pressure supports. As a corollary, we derive upper bounds for the Hausdorff dimensions of the free boundaries and show that the limiting free boundary has finite <span><math><mo>(</mo><mi>d</mi><mo>−</mo><mn>1</mn><mo>)</mo></math></span>-dimensional Hausdorff measure.</div></div>","PeriodicalId":51071,"journal":{"name":"Journal de Mathematiques Pures et Appliquees","volume":"203 ","pages":"Article 103752"},"PeriodicalIF":2.1,"publicationDate":"2025-06-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144320782","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}
Pub Date : 2025-06-16DOI: 10.1016/j.matpur.2025.103754
Min Ding , Huicheng Yin
Under the genuinely nonlinear assumption for 1-D strictly hyperbolic conservation laws, we investigate the geometric blowup of smooth solutions and the development of singularities when the small initial data fulfill the generic nondegenerate condition. At first, near the unique blowup point we give a precise description on the space-time blowup rate of the smooth solution and meanwhile derive the cusp singularity structure of characteristic envelope. These results are established through extending the smooth solution of the completely nonlinear blowup system across the blowup time. Subsequently, by utilizing a new form on the resulting 1-D strictly hyperbolic system with good components and one bad component, together with the choice of an efficient iterative scheme and some involved analyses, a weak entropy shock wave starting from the blowup point is constructed. As a byproduct, our result can be applied to the shock formation and construction for the 2-D supersonic steady compressible full Euler equations ( system), 1-D MHD equations ( system), 1-D elastic wave equations ( system) and 1-D full ideal compressible MHD equations ( system).
{"title":"Formation and construction of a shock wave for 1-D n × n strictly hyperbolic conservation laws with small smooth initial data","authors":"Min Ding , Huicheng Yin","doi":"10.1016/j.matpur.2025.103754","DOIUrl":"10.1016/j.matpur.2025.103754","url":null,"abstract":"<div><div>Under the genuinely nonlinear assumption for 1-D <span><math><mi>n</mi><mo>×</mo><mi>n</mi></math></span> strictly hyperbolic conservation laws, we investigate the geometric blowup of smooth solutions and the development of singularities when the small initial data fulfill the generic nondegenerate condition. At first, near the unique blowup point we give a precise description on the space-time blowup rate of the smooth solution and meanwhile derive the cusp singularity structure of characteristic envelope. These results are established through extending the smooth solution of the completely nonlinear blowup system across the blowup time. Subsequently, by utilizing a new form on the resulting 1-D strictly hyperbolic system with <span><math><mo>(</mo><mi>n</mi><mo>−</mo><mn>1</mn><mo>)</mo></math></span> good components and one bad component, together with the choice of an efficient iterative scheme and some involved analyses, a weak entropy shock wave starting from the blowup point is constructed. As a byproduct, our result can be applied to the shock formation and construction for the 2-D supersonic steady compressible full Euler equations (<span><math><mn>4</mn><mo>×</mo><mn>4</mn></math></span> system), 1-D MHD equations (<span><math><mn>5</mn><mo>×</mo><mn>5</mn></math></span> system), 1-D elastic wave equations (<span><math><mn>6</mn><mo>×</mo><mn>6</mn></math></span> system) and 1-D full ideal compressible MHD equations (<span><math><mn>7</mn><mo>×</mo><mn>7</mn></math></span> system).</div></div>","PeriodicalId":51071,"journal":{"name":"Journal de Mathematiques Pures et Appliquees","volume":"204 ","pages":"Article 103754"},"PeriodicalIF":2.1,"publicationDate":"2025-06-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144321251","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}
In this work we analyse the small-time reachability properties of a nonlinear parabolic equation, by means of a bilinear control, posed on a torus of arbitrary dimension d. Under a saturation hypothesis on the control operators, we show the small-time approximate controllability between states sharing the same sign. Moreover, in the one-dimensional case , we combine this property with a local exact controllability result, and prove the small-time exact controllability of any positive states towards the ground state of the evolution operator.
Dans ce travail, nous analysons les propriétés d'accessibilité en temps court d'une équation parabolique non linéaire, à l'aide d'un contrôle bilinéaire, posée sur un tore de dimension arbitraire d. Sous une hypothèse de saturation sur les opérateurs de contrôle, nous montrons la contrôlabilité approchée en temps court entre les états qui ont le même signe. De plus, dans le cas unidimensionnel , nous combinons cette propriété avec un résultat de contrôlabilité locale exacte, et prouvons la contrôlabilité exacte en temps court de tout état positif vers l'état fondamental de l'opérateur d'évolution.
{"title":"On the small-time bilinear control of a nonlinear heat equation: Global approximate controllability and exact controllability to trajectories","authors":"Alessandro Duca , Eugenio Pozzoli , Cristina Urbani","doi":"10.1016/j.matpur.2025.103758","DOIUrl":"10.1016/j.matpur.2025.103758","url":null,"abstract":"<div><div>In this work we analyse the small-time reachability properties of a nonlinear parabolic equation, by means of a bilinear control, posed on a torus of arbitrary dimension <em>d</em>. Under a saturation hypothesis on the control operators, we show the small-time approximate controllability between states sharing the same sign. Moreover, in the one-dimensional case <span><math><mi>d</mi><mo>=</mo><mn>1</mn></math></span>, we combine this property with a local exact controllability result, and prove the small-time exact controllability of any positive states towards the ground state of the evolution operator.</div><div>Dans ce travail, nous analysons les propriétés d'accessibilité en temps court d'une équation parabolique non linéaire, à l'aide d'un contrôle bilinéaire, posée sur un tore de dimension arbitraire <em>d</em>. Sous une hypothèse de saturation sur les opérateurs de contrôle, nous montrons la contrôlabilité approchée en temps court entre les états qui ont le même signe. De plus, dans le cas unidimensionnel <span><math><mi>d</mi><mo>=</mo><mn>1</mn></math></span>, nous combinons cette propriété avec un résultat de contrôlabilité locale exacte, et prouvons la contrôlabilité exacte en temps court de tout état positif vers l'état fondamental de l'opérateur d'évolution.</div></div>","PeriodicalId":51071,"journal":{"name":"Journal de Mathematiques Pures et Appliquees","volume":"203 ","pages":"Article 103758"},"PeriodicalIF":2.1,"publicationDate":"2025-06-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144320781","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}
Pub Date : 2025-06-16DOI: 10.1016/j.matpur.2025.103755
Giulio Ciraolo, Matteo Cozzi, Michele Gatti
A celebrated result by Gidas, Ni & Nirenberg asserts that positive classical solutions, decaying at infinity, to semilinear equations in must be radial and radially decreasing. In this paper, we consider both energy solutions in and non-energy local weak solutions to small perturbations of these equations, and study its quantitative stability counterpart.
To the best of our knowledge, the present work provides the first quantitative stability result for non-energy solutions to semilinear equations involving the Laplacian, even for the critical nonlinearity.
Gidas, Ni &；Nirenberg断言，在无穷远处衰减的半线性方程Δu+f(u)=0的正经典解在Rn中必须是径向和径向递减的。本文考虑了这些方程D1,2（Rn）的能量解和小扰动的非能量局部弱解，并研究了它们的定量稳定性对应项。据我们所知，目前的工作提供了第一个涉及拉普拉斯方程的半线性方程的非能量解的定量稳定性结果，甚至对于临界非线性也是如此。
{"title":"A quantitative study of radial symmetry for solutions to semilinear equations in Rn","authors":"Giulio Ciraolo, Matteo Cozzi, Michele Gatti","doi":"10.1016/j.matpur.2025.103755","DOIUrl":"10.1016/j.matpur.2025.103755","url":null,"abstract":"<div><div>A celebrated result by Gidas, Ni & Nirenberg asserts that positive classical solutions, decaying at infinity, to semilinear equations <span><math><mi>Δ</mi><mi>u</mi><mo>+</mo><mi>f</mi><mo>(</mo><mi>u</mi><mo>)</mo><mo>=</mo><mn>0</mn></math></span> in <span><math><msup><mrow><mi>R</mi></mrow><mrow><mi>n</mi></mrow></msup></math></span> must be radial and radially decreasing. In this paper, we consider both energy solutions in <span><math><msup><mrow><mi>D</mi></mrow><mrow><mn>1</mn><mo>,</mo><mn>2</mn></mrow></msup><mo>(</mo><msup><mrow><mi>R</mi></mrow><mrow><mi>n</mi></mrow></msup><mo>)</mo></math></span> and non-energy local weak solutions to small perturbations of these equations, and study its quantitative stability counterpart.</div><div>To the best of our knowledge, the present work provides the first quantitative stability result for non-energy solutions to semilinear equations involving the Laplacian, even for the critical nonlinearity.</div></div>","PeriodicalId":51071,"journal":{"name":"Journal de Mathematiques Pures et Appliquees","volume":"204 ","pages":"Article 103755"},"PeriodicalIF":2.1,"publicationDate":"2025-06-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144321252","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}
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