A bstract . We consider the modified Korteweg-de Vries equation (mKdV) and prove that given any sum 𝑃 of solitons and breathers of (mKdV) (with distinct velocities), there exists a solution 𝑝 of (mKdV) such that 𝑝 ( 𝑡 ) − 𝑃 ( 𝑡 ) → 0 when 𝑡 → +∞ , which we call multi-breather. In order to do this, we work at the 𝐻 2 level (even if usually solitons are considered at the 𝐻 1 level). We will show that this convergence takes place in any 𝐻 𝑠 space and that this convergence is exponentially fast in time. We also show that the constructed multi-breather is unique in two cases: in the class of solutions which converge to the profile 𝑃 faster than the inverse of a polynomial of a large enough degree in time (we will call this a super polynomial convergence), or (without hypothesis on the convergence rate), when all the velocities are positive.
{"title":"On the uniqueness of multi-breathers of the modified Korteweg–de Vries equation","authors":"A. Semenov","doi":"10.4171/rmi/1363","DOIUrl":"https://doi.org/10.4171/rmi/1363","url":null,"abstract":"A bstract . We consider the modified Korteweg-de Vries equation (mKdV) and prove that given any sum 𝑃 of solitons and breathers of (mKdV) (with distinct velocities), there exists a solution 𝑝 of (mKdV) such that 𝑝 ( 𝑡 ) − 𝑃 ( 𝑡 ) → 0 when 𝑡 → +∞ , which we call multi-breather. In order to do this, we work at the 𝐻 2 level (even if usually solitons are considered at the 𝐻 1 level). We will show that this convergence takes place in any 𝐻 𝑠 space and that this convergence is exponentially fast in time. We also show that the constructed multi-breather is unique in two cases: in the class of solutions which converge to the profile 𝑃 faster than the inverse of a polynomial of a large enough degree in time (we will call this a super polynomial convergence), or (without hypothesis on the convergence rate), when all the velocities are positive.","PeriodicalId":49604,"journal":{"name":"Revista Matematica Iberoamericana","volume":" ","pages":""},"PeriodicalIF":1.2,"publicationDate":"2021-09-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47705262","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}
We apply scattering theory on asymptotically hyperbolic manifolds to singular Yamabe metrics, applying the results to the study of the conformal geometry of compact manifolds with boundary. In particular, we define extrinsic versions of the conformally invariant powers of the Laplacian, or GJMS operators, on the boundary of any such manifold, along with associated extrinsic Q-curvatures. We use the existence and uniqueness of a singular Yamabe metric conformal to define also nonlocal extrinsic fractional GJMS operators on the boundary, and draw other global conclusions about the scattering operator, including a Gauss-Bonnet theorem in dimension four.
{"title":"Scattering on singular Yamabe spaces","authors":"S. Chang, Stephen E. McKeown, Paul Yang","doi":"10.4171/rmi/1390","DOIUrl":"https://doi.org/10.4171/rmi/1390","url":null,"abstract":"We apply scattering theory on asymptotically hyperbolic manifolds to singular Yamabe metrics, applying the results to the study of the conformal geometry of compact manifolds with boundary. In particular, we define extrinsic versions of the conformally invariant powers of the Laplacian, or GJMS operators, on the boundary of any such manifold, along with associated extrinsic Q-curvatures. We use the existence and uniqueness of a singular Yamabe metric conformal to define also nonlocal extrinsic fractional GJMS operators on the boundary, and draw other global conclusions about the scattering operator, including a Gauss-Bonnet theorem in dimension four.","PeriodicalId":49604,"journal":{"name":"Revista Matematica Iberoamericana","volume":" ","pages":""},"PeriodicalIF":1.2,"publicationDate":"2021-09-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42692752","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":"Hypersurfaces with prescribed curvatures in the de Sitter space","authors":"A. Roldán","doi":"10.4171/rmi/1279","DOIUrl":"https://doi.org/10.4171/rmi/1279","url":null,"abstract":"","PeriodicalId":49604,"journal":{"name":"Revista Matematica Iberoamericana","volume":" ","pages":""},"PeriodicalIF":1.2,"publicationDate":"2021-08-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42012609","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}
Francesco Paolo Maiale, Giorgio Tortone, B. Velichkov
This paper is dedicated to a free boundary system arising in the study of a class of shape optimization problems. The problem involves three variables: two functions $u$ and $v$, and a domain $Omega$; with $u$ and $v$ being both positive in $Omega$, vanishing simultaneously on $partialOmega$ and satisfying an overdetermined boundary value problem involving the product of their normal derivatives on $partialOmega$. Precisely, we consider solutions $u, v in C(B_1)$ of $$-Delta u= f quadtext{and} quad-Delta v=gquadtext{in}quad Omega={u>0}={v>0} ,qquad frac{partial u}{partial n}frac{partial v}{partial n}=Qquadtext{on}quad partialOmegacap B_1.$$ Our main result is an epsilon-regularity theorem for viscosity solutions of this free boundary system. We prove a partial Harnack inequality near flat points for the couple of auxiliary functions $sqrt{uv}$ and $frac12(u+v)$. Then, we use the gained space near the free boundary to transfer the improved flatness to the original solutions. Finally, using the partial Harnack inequality, we obtain an improvement-of-flatness result, which allows to conclude that flatness implies $C^{1,alpha}$ regularity.
本文研究了一类形状优化问题中出现的自由边界系统。该问题涉及三个变量:两个函数$u$和$v$，以及一个域$Omega$;$u$和$v$在$Omega$上都是正的，在$partialOmega$上同时消失，并且满足一个超定边值问题，涉及它们在$partialOmega$上的法向导数的乘积。确切地说，我们考虑$$-Delta u= f quadtext{and} quad-Delta v=gquadtext{in}quad Omega={u>0}={v>0} ,qquad frac{partial u}{partial n}frac{partial v}{partial n}=Qquadtext{on}quad partialOmegacap B_1.$$的解$u, v in C(B_1)$。我们的主要结果是该自由边界系统粘度解的一个ε -正则定理。我们证明了一对辅助函数$sqrt{uv}$和$frac12(u+v)$在平坦点附近的一个偏Harnack不等式。然后，利用获得的自由边界附近空间将改进后的平面度传递到原解中。最后，利用部分Harnack不等式，我们得到了平面性的改进结果，从而得出平面性隐含$C^{1,alpha}$正则性的结论。
{"title":"Epsilon-regularity for the solutions of a free boundary system","authors":"Francesco Paolo Maiale, Giorgio Tortone, B. Velichkov","doi":"10.4171/rmi/1430","DOIUrl":"https://doi.org/10.4171/rmi/1430","url":null,"abstract":"This paper is dedicated to a free boundary system arising in the study of a class of shape optimization problems. The problem involves three variables: two functions $u$ and $v$, and a domain $Omega$; with $u$ and $v$ being both positive in $Omega$, vanishing simultaneously on $partialOmega$ and satisfying an overdetermined boundary value problem involving the product of their normal derivatives on $partialOmega$. Precisely, we consider solutions $u, v in C(B_1)$ of $$-Delta u= f quadtext{and} quad-Delta v=gquadtext{in}quad Omega={u>0}={v>0} ,qquad frac{partial u}{partial n}frac{partial v}{partial n}=Qquadtext{on}quad partialOmegacap B_1.$$ Our main result is an epsilon-regularity theorem for viscosity solutions of this free boundary system. We prove a partial Harnack inequality near flat points for the couple of auxiliary functions $sqrt{uv}$ and $frac12(u+v)$. Then, we use the gained space near the free boundary to transfer the improved flatness to the original solutions. Finally, using the partial Harnack inequality, we obtain an improvement-of-flatness result, which allows to conclude that flatness implies $C^{1,alpha}$ regularity.","PeriodicalId":49604,"journal":{"name":"Revista Matematica Iberoamericana","volume":"1 1","pages":""},"PeriodicalIF":1.2,"publicationDate":"2021-08-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41665140","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}
Verena Bogelein, F. Duzaar, Naian Liao, Leah Schatzler
We demonstrate two proofs for the local H"older continuity of possibly sign-changing solutions to a class of doubly nonlinear parabolic equations whose prototype is [ partial_tbig(|u|^{q-1}ubig)-Delta_p u=0,quad p>2,quad 0
{"title":"On the Hölder regularity of signed solutions to a doubly nonlinear equation. Part II","authors":"Verena Bogelein, F. Duzaar, Naian Liao, Leah Schatzler","doi":"10.4171/rmi/1342","DOIUrl":"https://doi.org/10.4171/rmi/1342","url":null,"abstract":"We demonstrate two proofs for the local H\"older continuity of possibly sign-changing solutions to a class of doubly nonlinear parabolic equations whose prototype is [ partial_tbig(|u|^{q-1}ubig)-Delta_p u=0,quad p>2,quad 0<q<p-1. ] The first proof takes advantage of the expansion of positivity for the degenerate, parabolic $p$-Laplacian, thus simplifying the argument; whereas the other proof relies solely on the energy estimates for the doubly nonlinear parabolic equations. After proper adaptions of the interior arguments, we also obtain the boundary regularity for initial-boundary value problems of Dirichlet type and Neumann type.","PeriodicalId":49604,"journal":{"name":"Revista Matematica Iberoamericana","volume":" ","pages":""},"PeriodicalIF":1.2,"publicationDate":"2021-08-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46233406","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}
The elliptic sine-Gordon equation is a semilinear elliptic equation with a special double well potential. It has a family of explicit multiple-end solutions. We show that all the finite Morse index solutions belong to this family. It will also be proved that these solutions are nondegenerate, in the sense that the corresponding linearized operators have no nontrivial bounded kernel. Finally, we prove that the Morse index of 2n-end solutions is equal to n(n−1)/2.
{"title":"Classification of finite Morse index solutions to the elliptic sine-Gordon equation in the plane","authors":"Yong Liu, Juncheng Wei","doi":"10.4171/rmi/1296","DOIUrl":"https://doi.org/10.4171/rmi/1296","url":null,"abstract":"The elliptic sine-Gordon equation is a semilinear elliptic equation with a special double well potential. It has a family of explicit multiple-end solutions. We show that all the finite Morse index solutions belong to this family. It will also be proved that these solutions are nondegenerate, in the sense that the corresponding linearized operators have no nontrivial bounded kernel. Finally, we prove that the Morse index of 2n-end solutions is equal to n(n−1)/2.","PeriodicalId":49604,"journal":{"name":"Revista Matematica Iberoamericana","volume":"1 1","pages":""},"PeriodicalIF":1.2,"publicationDate":"2021-08-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70906558","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}
. It is shown that if γ : [ a,b ] → S 2 is C 3 with det( γ,γ ′ ,γ ′′ ) 6 = 0, and if A ⊆ R 3 is a Borel set, then dim π θ ( A ) ≥ min n 2 , dim A, dim A 2 + 34 o for a.e. θ ∈ [ a,b ], where π θ denotes projection onto the orthogonal complement of γ ( θ ) and “dim” refers to Hausdorff dimension. This partially resolves a conjecture of F¨assler and Orponen in the range 1 < dim A ≤ 3 / 2, which was previously known only for non-great circles. For 3 / 2 < dim A < 5 / 2 this improves the known lower bound for this problem.
{"title":"Restricted families of projections onto planes: The general case of nonvanishing geodesic curvature","authors":"Terence L. J. Harris","doi":"10.4171/RMI/1387","DOIUrl":"https://doi.org/10.4171/RMI/1387","url":null,"abstract":". It is shown that if γ : [ a,b ] → S 2 is C 3 with det( γ,γ ′ ,γ ′′ ) 6 = 0, and if A ⊆ R 3 is a Borel set, then dim π θ ( A ) ≥ min n 2 , dim A, dim A 2 + 34 o for a.e. θ ∈ [ a,b ], where π θ denotes projection onto the orthogonal complement of γ ( θ ) and “dim” refers to Hausdorff dimension. This partially resolves a conjecture of F¨assler and Orponen in the range 1 < dim A ≤ 3 / 2, which was previously known only for non-great circles. For 3 / 2 < dim A < 5 / 2 this improves the known lower bound for this problem.","PeriodicalId":49604,"journal":{"name":"Revista Matematica Iberoamericana","volume":" ","pages":""},"PeriodicalIF":1.2,"publicationDate":"2021-07-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42248226","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}
I. P. C. E. Silva, J. Flores, Kledilson P. R. Honorato
We revisit certain path-lifting and path-continuation properties of abstract maps as described in the work of F. Browder and R. Rheindboldt in 1950-1960s, and apply their elegant theory to exponential maps. We obtain thereby a number of novel results of existence and multiplicity of geodesics joining any two points of a connected affine manifold, as well as causal geodesics connecting any two causally related points on a Lorentzian manifold. These results include a generalization of the well-known Hadamard-Cartan theorem of Riemannian geometry to the affine manifold context, as well as a new version of the so-called Lorentzian Hadamard-Cartan theorem using weaker assumptions than global hyperbolicity and timelike 1-connectedness required in the extant version. We also include a general discription of pseudoconvexity and disprisonment of broad classes of geodesics in terms of suitable restrictions of the exponential map. The latter description sheds further light on the the relation between pseudoconvexity and disprisonment of a given such class on the one hand, and geodesic connectedness by members of that class on the other.
{"title":"Path-lifting properties of the exponential map with applications to geodesics","authors":"I. P. C. E. Silva, J. Flores, Kledilson P. R. Honorato","doi":"10.4171/rmi/1364","DOIUrl":"https://doi.org/10.4171/rmi/1364","url":null,"abstract":"We revisit certain path-lifting and path-continuation properties of abstract maps as described in the work of F. Browder and R. Rheindboldt in 1950-1960s, and apply their elegant theory to exponential maps. We obtain thereby a number of novel results of existence and multiplicity of geodesics joining any two points of a connected affine manifold, as well as causal geodesics connecting any two causally related points on a Lorentzian manifold. These results include a generalization of the well-known Hadamard-Cartan theorem of Riemannian geometry to the affine manifold context, as well as a new version of the so-called Lorentzian Hadamard-Cartan theorem using weaker assumptions than global hyperbolicity and timelike 1-connectedness required in the extant version. We also include a general discription of pseudoconvexity and disprisonment of broad classes of geodesics in terms of suitable restrictions of the exponential map. The latter description sheds further light on the the relation between pseudoconvexity and disprisonment of a given such class on the one hand, and geodesic connectedness by members of that class on the other.","PeriodicalId":49604,"journal":{"name":"Revista Matematica Iberoamericana","volume":" ","pages":""},"PeriodicalIF":1.2,"publicationDate":"2021-07-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46148702","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}
We prove sharp bounds for the size of superlevel sets {x ∈ R : |f(x)| > α} where α > 0 and f : R → C is a Schwartz function with Fourier transform supported in an R-neighborhood of the truncated parabola P. These estimates imply the small cap decoupling theorem for P from [DGW20] and the canonical decoupling theorem for P from [BD15]. New (l, L) small cap decoupling inequalities also follow from our sharp level set estimates. In this paper, we further develop the high/low frequency proof of decoupling for the parabola [GMW20] to prove sharp level set estimates which recover and refine the small cap decoupling results for the parabola in [DGW20]. We begin by describing the problem and our results in terms of exponential sums. The main results in full generality are in §1. For N ≥ 1, R ∈ [N,N2], and 2 ≤ p, let D(N,R, p) denote the smallest constant so that
{"title":"Sharp superlevel set estimates for small cap decouplings of the parabola","authors":"Yu Fu, L. Guth, Dominique Maldague","doi":"10.4171/rmi/1393","DOIUrl":"https://doi.org/10.4171/rmi/1393","url":null,"abstract":"We prove sharp bounds for the size of superlevel sets {x ∈ R : |f(x)| > α} where α > 0 and f : R → C is a Schwartz function with Fourier transform supported in an R-neighborhood of the truncated parabola P. These estimates imply the small cap decoupling theorem for P from [DGW20] and the canonical decoupling theorem for P from [BD15]. New (l, L) small cap decoupling inequalities also follow from our sharp level set estimates. In this paper, we further develop the high/low frequency proof of decoupling for the parabola [GMW20] to prove sharp level set estimates which recover and refine the small cap decoupling results for the parabola in [DGW20]. We begin by describing the problem and our results in terms of exponential sums. The main results in full generality are in §1. For N ≥ 1, R ∈ [N,N2], and 2 ≤ p, let D(N,R, p) denote the smallest constant so that","PeriodicalId":49604,"journal":{"name":"Revista Matematica Iberoamericana","volume":" ","pages":""},"PeriodicalIF":1.2,"publicationDate":"2021-07-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43122309","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}
. In this paper, we establish curvature estimates for p -convex hypersurfaces in R n +1 of prescribed curvature with p ≥ n 2 . The existence of a star-shaped hypersurface of prescribed curvature is obtained. We also prove a type of interior C 2 estimates for solutions to the Dirichlet problem of the corresponding equation. Mathematical Subject Classification (2010): 53C42; 53C21; 35J60.
{"title":"Curvature estimates for $p$-convex hypersurfaces of prescribed curvature","authors":"Weisong Dong","doi":"10.4171/rmi/1348","DOIUrl":"https://doi.org/10.4171/rmi/1348","url":null,"abstract":". In this paper, we establish curvature estimates for p -convex hypersurfaces in R n +1 of prescribed curvature with p ≥ n 2 . The existence of a star-shaped hypersurface of prescribed curvature is obtained. We also prove a type of interior C 2 estimates for solutions to the Dirichlet problem of the corresponding equation. Mathematical Subject Classification (2010): 53C42; 53C21; 35J60.","PeriodicalId":49604,"journal":{"name":"Revista Matematica Iberoamericana","volume":" ","pages":""},"PeriodicalIF":1.2,"publicationDate":"2021-07-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45292513","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}
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