Pub Date : 2025-12-01Epub Date: 2025-09-30DOI: 10.1016/j.matpur.2025.103804
Nathanaël Boutillon
We consider a nonlocal Fisher-KPP equation that models a population structured in space and in phenotype. The population lives in a heterogeneous periodic environment: the diffusion coefficient, the mutation coefficient and the fitness of an individual may depend on its spatial position and on its phenotype.
We first prove a Freidlin-Gärtner formula for the spreading speed of the population. We then study the behaviour of the spreading speed in different scaling limits (small and large period, small and large mutation coefficient). Finally, we exhibit new phenomena arising thanks to the phenotypic dimension.
Our results are also valid when the phenotype is seen as another spatial variable along which the population does not spread.
{"title":"Qualitative properties of the spreading speed of a population structured in space and in phenotype","authors":"Nathanaël Boutillon","doi":"10.1016/j.matpur.2025.103804","DOIUrl":"10.1016/j.matpur.2025.103804","url":null,"abstract":"<div><div>We consider a nonlocal Fisher-KPP equation that models a population structured in space and in phenotype. The population lives in a heterogeneous periodic environment: the diffusion coefficient, the mutation coefficient and the fitness of an individual may depend on its spatial position and on its phenotype.</div><div>We first prove a Freidlin-Gärtner formula for the spreading speed of the population. We then study the behaviour of the spreading speed in different scaling limits (small and large period, small and large mutation coefficient). Finally, we exhibit new phenomena arising thanks to the phenotypic dimension.</div><div>Our results are also valid when the phenotype is seen as another spatial variable along which the population does not spread.</div></div>","PeriodicalId":51071,"journal":{"name":"Journal de Mathematiques Pures et Appliquees","volume":"204 ","pages":"Article 103804"},"PeriodicalIF":2.3,"publicationDate":"2025-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145264963","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 : 2025-12-01Epub Date: 2025-09-12DOI: 10.1016/j.matpur.2025.103794
Baohua Fu , Jie Liu
The algebra of symmetric tensors of a projective manifold X leads to a natural dominant affinization morphism It is shown that is birational if and only if is big. We prove that if is birational, then is a symplectic variety endowed with the Schouten–Nijenhuis bracket if and only if is of Fano type, which is the case for smooth projective toric varieties, smooth horospherical varieties with small boundary, and the quintic del Pezzo threefold. These give examples of a distinguished class of conical symplectic varieties, which we call symplectic orbifold cones.
对对称张量S(X)的代数，其中对投影流形X的H0（X,S•TX）是一个自然优势仿射态φX:T X ZX是一个自然优势仿射态。证明了φX当且仅当TX较大时是两位数的。证明了φX是双分型的，则当且仅当PTX为Fano型时，ZX是赋有Schouten-Nijenhuis括弧的简型变种，对于光滑投影环型变种、小边界光滑全球型变种和五次del Pezzo三重型都是如此。这些给出了一类特殊的圆锥辛变异体的例子，我们称之为辛轨道锥。
{"title":"Symplectic singularities arising from algebras of symmetric tensors","authors":"Baohua Fu , Jie Liu","doi":"10.1016/j.matpur.2025.103794","DOIUrl":"10.1016/j.matpur.2025.103794","url":null,"abstract":"<div><div>The algebra of symmetric tensors <span><math><mi>S</mi><mo>(</mo><mi>X</mi><mo>)</mo><mo>≔</mo><msup><mrow><mi>H</mi></mrow><mrow><mn>0</mn></mrow></msup><mo>(</mo><mi>X</mi><mo>,</mo><msup><mrow><mi>S</mi></mrow><mrow><mo>•</mo></mrow></msup><msub><mrow><mi>T</mi></mrow><mrow><mi>X</mi></mrow></msub><mo>)</mo></math></span> of a projective manifold <em>X</em> leads to a natural dominant affinization morphism<span><span><span><math><msub><mrow><mi>φ</mi></mrow><mrow><mi>X</mi></mrow></msub><mo>:</mo><msup><mrow><mi>T</mi></mrow><mrow><mo>⁎</mo></mrow></msup><mi>X</mi><mo>⟶</mo><msub><mrow><mi>Z</mi></mrow><mrow><mi>X</mi></mrow></msub><mo>≔</mo><mi>Spec</mi><mspace></mspace><mi>S</mi><mo>(</mo><mi>X</mi><mo>)</mo><mo>.</mo></math></span></span></span> It is shown that <span><math><msub><mrow><mi>φ</mi></mrow><mrow><mi>X</mi></mrow></msub></math></span> is birational if and only if <span><math><msub><mrow><mi>T</mi></mrow><mrow><mi>X</mi></mrow></msub></math></span> is big. We prove that if <span><math><msub><mrow><mi>φ</mi></mrow><mrow><mi>X</mi></mrow></msub></math></span> is birational, then <span><math><msub><mrow><mi>Z</mi></mrow><mrow><mi>X</mi></mrow></msub></math></span> is a symplectic variety endowed with the Schouten–Nijenhuis bracket if and only if <span><math><mi>P</mi><msub><mrow><mi>T</mi></mrow><mrow><mi>X</mi></mrow></msub></math></span> is of Fano type, which is the case for smooth projective toric varieties, smooth horospherical varieties with small boundary, and the quintic del Pezzo threefold. These give examples of a distinguished class of conical symplectic varieties, which we call symplectic orbifold cones.</div></div>","PeriodicalId":51071,"journal":{"name":"Journal de Mathematiques Pures et Appliquees","volume":"204 ","pages":"Article 103794"},"PeriodicalIF":2.3,"publicationDate":"2025-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145109260","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 : 2025-12-01Epub Date: 2025-09-12DOI: 10.1016/j.matpur.2025.103793
Massimiliano Mella
Two projective varieties are said to be Cremona equivalent if there is a Cremona modification sending one onto the other. In the last decade, Cremona equivalence has been investigated widely, and we now have a complete theory for non-divisorial reduced schemes. The case of irreducible divisors is completely different, and not much is known besides the case of plane curves and a few classes of surfaces. In particular, for plane curves it is a classical result that an irreducible plane curve is Cremona equivalent to a line if and only if its log-Kodaira dimension is negative. This can be interpreted as the log version of Castelnuovo's rationality criterion for surfaces. One expects that a similar result for surfaces in projective space should not be true, as it is false, the generalization in higher dimensions of Castelnuovo's Rationality Theorem. In this paper, the first example of such behavior is provided, exhibiting a rational surface in the projective space with negative log-Kodaira dimension, which is not Cremona equivalent to a plane. This can be thought of as a sort of log Iskovkikh-Manin, Clemens-Griffith, Artin-Mumford example. Using this example, it is then possible to show that Cremona equivalence to a plane is neither open nor closed among log pairs with negative Kodaira dimension.
{"title":"Cremona equivalence and log Kodaira dimension","authors":"Massimiliano Mella","doi":"10.1016/j.matpur.2025.103793","DOIUrl":"10.1016/j.matpur.2025.103793","url":null,"abstract":"<div><div>Two projective varieties are said to be Cremona equivalent if there is a Cremona modification sending one onto the other. In the last decade, Cremona equivalence has been investigated widely, and we now have a complete theory for non-divisorial reduced schemes. The case of irreducible divisors is completely different, and not much is known besides the case of plane curves and a few classes of surfaces. In particular, for plane curves it is a classical result that an irreducible plane curve is Cremona equivalent to a line if and only if its log-Kodaira dimension is negative. This can be interpreted as the log version of Castelnuovo's rationality criterion for surfaces. One expects that a similar result for surfaces in projective space should not be true, as it is false, the generalization in higher dimensions of Castelnuovo's Rationality Theorem. In this paper, the first example of such behavior is provided, exhibiting a rational surface in the projective space with negative log-Kodaira dimension, which is not Cremona equivalent to a plane. This can be thought of as a sort of log Iskovkikh-Manin, Clemens-Griffith, Artin-Mumford example. Using this example, it is then possible to show that Cremona equivalence to a plane is neither open nor closed among log pairs with negative Kodaira dimension.</div></div>","PeriodicalId":51071,"journal":{"name":"Journal de Mathematiques Pures et Appliquees","volume":"204 ","pages":"Article 103793"},"PeriodicalIF":2.3,"publicationDate":"2025-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145104686","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 : 2025-12-01Epub Date: 2025-08-28DOI: 10.1016/j.matpur.2025.103784
Giorgio Poggesi
The quantitative analysis of bubbling phenomena for almost constant mean curvature boundaries is an important question having significant applications in various fields including capillarity theory and the study of mean curvature flows. Such a quantitative analysis was initiated in Ciraolo and Maggi (2017) [3], where the first quantitative result of proximity to a set of disjoint balls of equal radii was obtained in terms of a uniform deviation of the mean curvature from being constant. Weakening the measure of the deviation in such a result is a delicate issue that is crucial in view of the applications for mean curvature flows. Some progress in this direction was recently made in Julin and Niinikoski (2023) [12], where -deviations are considered for domains in . In the present paper we significantly weaken the measure of the deviation, obtaining a quantitative result of proximity to a set of disjoint balls of equal radii for the following deviation which is clearly even weaker than .
{"title":"Bubbling and quantitative stability for Alexandrov's Soap Bubble Theorem with L1-type deviations","authors":"Giorgio Poggesi","doi":"10.1016/j.matpur.2025.103784","DOIUrl":"10.1016/j.matpur.2025.103784","url":null,"abstract":"<div><div>The quantitative analysis of bubbling phenomena for almost constant mean curvature boundaries is an important question having significant applications in various fields including capillarity theory and the study of mean curvature flows. Such a quantitative analysis was initiated in Ciraolo and Maggi (2017) <span><span>[3]</span></span>, where the first quantitative result of proximity to a set of disjoint balls of equal radii was obtained in terms of a uniform deviation of the mean curvature from being constant. Weakening the measure of the deviation in such a result is a delicate issue that is crucial in view of the applications for mean curvature flows. Some progress in this direction was recently made in Julin and Niinikoski (2023) <span><span>[12]</span></span>, where <span><math><msup><mrow><mi>L</mi></mrow><mrow><mi>N</mi><mo>−</mo><mn>1</mn></mrow></msup></math></span>-deviations are considered for domains in <span><math><msup><mrow><mi>R</mi></mrow><mrow><mi>N</mi></mrow></msup></math></span>. In the present paper we significantly weaken the measure of the deviation, obtaining a quantitative result of proximity to a set of disjoint balls of equal radii for the following deviation<span><span><span><math><munder><mo>∫</mo><mrow><mo>∂</mo><mi>Ω</mi></mrow></munder><msup><mrow><mo>(</mo><msub><mrow><mi>H</mi></mrow><mrow><mn>0</mn></mrow></msub><mo>−</mo><mi>H</mi><mo>)</mo></mrow><mrow><mo>+</mo></mrow></msup><mi>d</mi><msub><mrow><mi>S</mi></mrow><mrow><mi>x</mi></mrow></msub><mo>,</mo><mspace></mspace><mtext> where </mtext><mrow><mo>{</mo><mtable><mtr><mtd><mi>H</mi><mtext> is the mean curvature of </mtext><mo>∂</mo><mi>Ω</mi><mo>,</mo></mtd></mtr><mtr><mtd><msub><mrow><mi>H</mi></mrow><mrow><mn>0</mn></mrow></msub><mo>:</mo><mo>=</mo><mfrac><mrow><mo>|</mo><mo>∂</mo><mi>Ω</mi><mo>|</mo></mrow><mrow><mi>N</mi><mo>|</mo><mi>Ω</mi><mo>|</mo></mrow></mfrac><mo>,</mo></mtd></mtr><mtr><mtd><msup><mrow><mo>(</mo><msub><mrow><mi>H</mi></mrow><mrow><mn>0</mn></mrow></msub><mo>−</mo><mi>H</mi><mo>)</mo></mrow><mrow><mo>+</mo></mrow></msup><mo>:</mo><mo>=</mo><mi>max</mi><mo></mo><mrow><mo>{</mo><msub><mrow><mi>H</mi></mrow><mrow><mn>0</mn></mrow></msub><mo>−</mo><mi>H</mi><mo>,</mo><mn>0</mn><mo>}</mo></mrow><mo>,</mo></mtd></mtr></mtable></mrow></math></span></span></span> which is clearly even weaker than <span><math><msub><mrow><mo>‖</mo><msub><mrow><mi>H</mi></mrow><mrow><mn>0</mn></mrow></msub><mo>−</mo><mi>H</mi><mo>‖</mo></mrow><mrow><msup><mrow><mi>L</mi></mrow><mrow><mn>1</mn></mrow></msup><mo>(</mo><mo>∂</mo><mi>Ω</mi><mo>)</mo></mrow></msub></math></span>.</div></div>","PeriodicalId":51071,"journal":{"name":"Journal de Mathematiques Pures et Appliquees","volume":"204 ","pages":"Article 103784"},"PeriodicalIF":2.3,"publicationDate":"2025-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145009927","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 : 2025-12-01Epub Date: 2025-09-30DOI: 10.1016/j.matpur.2025.103806
Juhi Jang , Jiaqi Liu , Nader Masmoudi
In this article, we construct a continuum family of self-similar waiting time solutions for the one-dimensional compressible Euler equations for the adiabatic exponent in the half-line with the vacuum boundary. The solutions are confined by a stationary vacuum interface for a finite time with at least regularity of the velocity and the sound speed up to the boundary. Subsequently, the solutions undergo the change of the behavior, becoming only Hölder continuous near the singular point, and simultaneously transition to the solutions to the vacuum moving boundary Euler equations satisfying the physical vacuum condition. When the boundary starts moving, a weak discontinuity emanating from the singular point along the sonic curve emerges. The solutions are locally smooth in the interior region away from the vacuum boundary and the sonic curve.
{"title":"Waiting time solutions in gas dynamics","authors":"Juhi Jang , Jiaqi Liu , Nader Masmoudi","doi":"10.1016/j.matpur.2025.103806","DOIUrl":"10.1016/j.matpur.2025.103806","url":null,"abstract":"<div><div>In this article, we construct a continuum family of self-similar waiting time solutions for the one-dimensional compressible Euler equations for the adiabatic exponent <span><math><mi>γ</mi><mo>∈</mo><mo>(</mo><mn>1</mn><mo>,</mo><mn>3</mn><mo>)</mo></math></span> in the half-line with the vacuum boundary. The solutions are confined by a stationary vacuum interface for a finite time with at least <span><math><msup><mrow><mi>C</mi></mrow><mrow><mn>1</mn></mrow></msup></math></span> regularity of the velocity and the sound speed up to the boundary. Subsequently, the solutions undergo the change of the behavior, becoming only Hölder continuous near the singular point, and simultaneously transition to the solutions to the vacuum moving boundary Euler equations satisfying the physical vacuum condition. When the boundary starts moving, a weak discontinuity emanating from the singular point along the sonic curve emerges. The solutions are locally smooth in the interior region away from the vacuum boundary and the sonic curve.</div></div>","PeriodicalId":51071,"journal":{"name":"Journal de Mathematiques Pures et Appliquees","volume":"204 ","pages":"Article 103806"},"PeriodicalIF":2.3,"publicationDate":"2025-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145264964","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 : 2025-12-01Epub 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":"2025-12-01","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}
Pub Date : 2025-12-01Epub Date: 2025-07-21DOI: 10.1016/j.matpur.2025.103776
Margherita Solci
We develop a general approach, using local interpolation inequalities, to non-convex integral functionals depending on the gradient with a singular perturbation by derivatives of order . When applied to functionals giving rise to free-discontinuity energies, such methods permit to change boundary values for derivatives up to order in problems defining density functions for the jump part, thus allowing to prove optimal-profile formulas, and to deduce compactness and lower bounds. As an application, we prove that for k-th order perturbations of energies depending on the gradient behaving as a constant at infinity, the jump energy density is a constant times the k-th root of the jump size. The result is first proved for truncated quadratic energy densities and in the one-dimensional case, from which the general higher-dimensional case can be obtained by slicing techniques. A wide class of non-convex energies can be studied as an envelope of these particular ones. Finally, we remark that an approximation of the Mumford–Shah functional can be obtained by letting k tend to infinity. We also derive a new approximation of the Blake-Zisserman functional.
{"title":"Local interpolation techniques for higher-order singular perturbations of non-convex functionals: Free-discontinuity problems","authors":"Margherita Solci","doi":"10.1016/j.matpur.2025.103776","DOIUrl":"10.1016/j.matpur.2025.103776","url":null,"abstract":"<div><div>We develop a general approach, using local interpolation inequalities, to non-convex integral functionals depending on the gradient with a singular perturbation by derivatives of order <span><math><mi>k</mi><mo>≥</mo><mn>2</mn></math></span>. When applied to functionals giving rise to free-discontinuity energies, such methods permit to change boundary values for derivatives up to order <span><math><mi>k</mi><mo>−</mo><mn>1</mn></math></span> in problems defining density functions for the jump part, thus allowing to prove optimal-profile formulas, and to deduce compactness and lower bounds. As an application, we prove that for <em>k</em>-th order perturbations of energies depending on the gradient behaving as a constant at infinity, the jump energy density is a constant <span><math><msub><mrow><mi>m</mi></mrow><mrow><mi>k</mi></mrow></msub></math></span> times the <em>k</em>-th root of the jump size. The result is first proved for truncated quadratic energy densities and in the one-dimensional case, from which the general higher-dimensional case can be obtained by slicing techniques. A wide class of non-convex energies can be studied as an envelope of these particular ones. Finally, we remark that an approximation of the Mumford–Shah functional can be obtained by letting <em>k</em> tend to infinity. We also derive a new approximation of the Blake-Zisserman functional.</div></div>","PeriodicalId":51071,"journal":{"name":"Journal de Mathematiques Pures et Appliquees","volume":"204 ","pages":"Article 103776"},"PeriodicalIF":2.1,"publicationDate":"2025-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144702622","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 : 2025-12-01Epub 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":"2025-12-01","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}
Pub Date : 2025-12-01Epub Date: 2025-06-13DOI: 10.1016/j.matpur.2025.103761
Jean-Michel Coron , Shengquan Xiang
In this paper, we study the controllability and stabilization problems of the harmonic map heat flow from a circle to a sphere. Combining ideas from control theory, heat flow, differential geometry, and asymptotic analysis, we obtain several important properties, such as small-time local controllability, local quantitative rapid stabilization, obstruction to semi-global asymptotic stabilization, and global controllability to geodesics. Surprisingly, due to the geometric feature of the equation we can also prove the small-time global controllability between harmonic maps within the same homotopy class for general compact Riemannian manifold targets, which is to be compared with the analogous but longstanding open problem for nonlinear heat equations.
{"title":"Global controllability to harmonic maps of the heat flow from a circle to a sphere","authors":"Jean-Michel Coron , Shengquan Xiang","doi":"10.1016/j.matpur.2025.103761","DOIUrl":"10.1016/j.matpur.2025.103761","url":null,"abstract":"<div><div>In this paper, we study the controllability and stabilization problems of the harmonic map heat flow from a circle to a sphere. Combining ideas from control theory, heat flow, differential geometry, and asymptotic analysis, we obtain several important properties, such as small-time local controllability, local quantitative rapid stabilization, obstruction to semi-global asymptotic stabilization, and global controllability to geodesics. Surprisingly, due to the geometric feature of the equation we can also prove the small-time global controllability between harmonic maps within the same homotopy class for general compact Riemannian manifold targets, which is to be compared with the analogous but longstanding open problem for nonlinear heat equations.</div></div>","PeriodicalId":51071,"journal":{"name":"Journal de Mathematiques Pures et Appliquees","volume":"204 ","pages":"Article 103761"},"PeriodicalIF":2.1,"publicationDate":"2025-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144491518","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 : 2025-12-01Epub Date: 2025-09-30DOI: 10.1016/j.matpur.2025.103802
Guofang Wang , Chao Xia , Xuwen Zhang
In this paper, we establish monotonicity formulas for capillary surfaces in the half-space and in the unit ball and extend the result of Volkmann (2016) [27] for surfaces with free boundary. As applications, we obtain Li-Yau-type inequalities for the Willmore energy of capillary surfaces, and extend Fraser-Schoen's optimal area estimate for minimal free boundary surfaces in (2011) [10] to the capillary setting, which is different to another optimal area estimate proved by Brendle (2023) [5].
{"title":"Monotonicity formulas for capillary surfaces","authors":"Guofang Wang , Chao Xia , Xuwen Zhang","doi":"10.1016/j.matpur.2025.103802","DOIUrl":"10.1016/j.matpur.2025.103802","url":null,"abstract":"<div><div>In this paper, we establish monotonicity formulas for capillary surfaces in the half-space <span><math><msubsup><mrow><mi>R</mi></mrow><mrow><mo>+</mo></mrow><mrow><mn>3</mn></mrow></msubsup></math></span> and in the unit ball <span><math><msup><mrow><mi>B</mi></mrow><mrow><mn>3</mn></mrow></msup></math></span> and extend the result of Volkmann (2016) <span><span>[27]</span></span> for surfaces with free boundary. As applications, we obtain Li-Yau-type inequalities for the Willmore energy of capillary surfaces, and extend Fraser-Schoen's optimal area estimate for minimal free boundary surfaces in <span><math><msup><mrow><mi>B</mi></mrow><mrow><mn>3</mn></mrow></msup></math></span> (2011) <span><span>[10]</span></span> to the capillary setting, which is different to another optimal area estimate proved by Brendle (2023) <span><span>[5]</span></span>.</div></div>","PeriodicalId":51071,"journal":{"name":"Journal de Mathematiques Pures et Appliquees","volume":"204 ","pages":"Article 103802"},"PeriodicalIF":2.3,"publicationDate":"2025-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145219417","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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