Abstract We are interested in sharp functional inequalities for the coherent state transform related to the Wehrl conjecture and its generalizations. This conjecture was settled by Lieb in the case of the Heisenberg group, Lieb and Solovej for SU(2), and Kulikov for SU(1, 1) and the affine group. In this article, we give alternative proofs and characterize, for the first time, the optimizers in the general case. We also extend the recent Faber-Krahn-type inequality for Heisenberg coherent states, due to Nicola and Tilli, to the SU(2) and SU(1, 1) cases. Finally, we prove a family of reverse Hölder inequalities for polynomials, conjectured by Bodmann.
{"title":"Sharp inequalities for coherent states and their optimizers","authors":"R. Frank","doi":"10.1515/ans-2022-0050","DOIUrl":"https://doi.org/10.1515/ans-2022-0050","url":null,"abstract":"Abstract We are interested in sharp functional inequalities for the coherent state transform related to the Wehrl conjecture and its generalizations. This conjecture was settled by Lieb in the case of the Heisenberg group, Lieb and Solovej for SU(2), and Kulikov for SU(1, 1) and the affine group. In this article, we give alternative proofs and characterize, for the first time, the optimizers in the general case. We also extend the recent Faber-Krahn-type inequality for Heisenberg coherent states, due to Nicola and Tilli, to the SU(2) and SU(1, 1) cases. Finally, we prove a family of reverse Hölder inequalities for polynomials, conjectured by Bodmann.","PeriodicalId":7191,"journal":{"name":"Advanced Nonlinear Studies","volume":" ","pages":""},"PeriodicalIF":1.8,"publicationDate":"2022-10-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49650062","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}
Abstract We establish the Caccioppoli inequality, a reverse Hölder inequality in the spirit of the classic estimate of Meyers, and construct the fundamental solution for linear elliptic differential equations of order 2 m 2m with certain lower order terms.
{"title":"Gradient estimates and the fundamental solution for higher-order elliptic systems with lower-order terms","authors":"Ariel Barton, M. J. Duffy","doi":"10.1515/ans-2022-0064","DOIUrl":"https://doi.org/10.1515/ans-2022-0064","url":null,"abstract":"Abstract We establish the Caccioppoli inequality, a reverse Hölder inequality in the spirit of the classic estimate of Meyers, and construct the fundamental solution for linear elliptic differential equations of order 2 m 2m with certain lower order terms.","PeriodicalId":7191,"journal":{"name":"Advanced Nonlinear Studies","volume":" ","pages":""},"PeriodicalIF":1.8,"publicationDate":"2022-10-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49019931","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}
Abstract We consider the existence and nonexistence of the positive solution for the following Brézis-Nirenberg problem with logarithmic perturbation: − Δ u = ∣ u ∣ 2 ∗ − 2 u + λ u + μ u log u 2 x ∈ Ω , u = 0 x ∈ ∂ Ω , left{phantom{rule[-1.25em]{}{0ex}}begin{array}{ll}-Delta u={| u| }^{{2}^{ast }-2}u+lambda u+mu ulog {u}^{2}hspace{1.0em}& xin Omega , u=0hspace{1.0em}& xin partial Omega ,end{array}right. where Ω ⊂ R N Omega subset {{mathbb{R}}}^{N} is a bounded open domain, λ , μ ∈ R lambda ,mu in {mathbb{R}} , N ≥ 3 Nge 3 and 2 ∗ ≔ 2 N N − 2 {2}^{ast }:= frac{2N}{N-2} is the critical Sobolev exponent for the embedding H 0 1 ( Ω ) ↪ L 2 ∗ ( Ω ) {H}_{0}^{1}left(Omega )hspace{0.33em}hookrightarrow hspace{0.33em}{L}^{{2}^{ast }}left(Omega ) . The uncertainty of the sign of s log s 2 slog {s}^{2} in ( 0 , + ∞ ) left(0,+infty ) has some interest in itself. We will show the existence of positive ground state solution, which is of mountain pass type provided λ ∈ R , μ > 0 lambda in {mathbb{R}},mu gt 0 and N ≥ 4 Nge 4 . While the case of μ < 0 mu lt 0 is thornier. However, for N = 3 , 4 N=3,4 , λ ∈ ( − ∞ , λ 1 ( Ω ) ) lambda in left(-infty ,{lambda }_{1}left(Omega )) , we can also establish the existence of positive solution under some further suitable assumptions. A nonexistence result is also obtained for μ < 0 mu lt 0 and − ( N − 2 ) μ 2 + ( N − 2 ) μ 2 log − ( N − 2 ) μ 2 + λ − λ 1 ( Ω ) ≥ 0 -frac{left(N-2)mu }{2}+frac{left(N-2)mu }{2}log left(-frac{left(N-2)mu }{2}right)+lambda -{lambda }_{1}left(Omega )ge 0 if N ≥ 3 Nge 3 . Comparing with the results in the study by Brézis and Nirenberg (Positive solutions of nonlinear elliptic equations involving critical Sobolev exponents, Comm. Pure Appl. Math. 36 (1983), 437–477), some new interesting phenomenon occurs when the parameter μ mu on logarithmic perturbation is not zero.
摘要我们考虑了下述具有对数扰动的br - nirenberg问题的正解的存在性和不存在性:−Δ u =∣u∣2∗−2 u + λ u + μ u log u 2 x∈Ω， u = 0 x∈∂Ω， left ｛phantom{rule[-1.25em]{}{0ex}}begin{array}{ll}-Delta u={| u| }^{{2}^{ast }-2}u+lambda u+mu ulog {u}^{2}hspace{1.0em}& xin Omega , u=0hspace{1.0em}& xin partial Omega ,end{array}right。其中Ω∧R N Omegasubset{{mathbb{R}}} ^{N}是有界开放域，λ， μ∈R lambda, muin{mathbb{R}}, N≥3 N ge 3和2∗其中，N−{2}^ {ast:=}frac{2N}{N-2}是嵌入h＿1 (Ω)“L 2∗(Ω) {H＿0}^{1}{}left (Omega) hspace{0.33em}hookrightarrow ^2hspace{0.33em}{L}^{{}{ast}}left (Omega)”的临界Sobolev指数。s log s 2 s log s{^}2{ in(0，+∞)}left (0，+ infty)的符号的不确定性本身就很有趣。我们将证明在λ∈R， μ > 0 lambdain{mathbb{R}}, mugt 0和N≥4 N ge 4的条件下存在山口型正基态解。而μ < 0 mult 0的情况则比较棘手。然而，对于N=3,4 N=3,4， λ∈(−∞，λ 1 (Ω)) lambdainleft (- infty, {lambda ＿1}{}left (Omega))，我们还可以在进一步适当的假设下建立正解的存在性。当N≥3 N ge 3时，得到了μ < 0 mult 0和−(N−2)μ 2 + (N−2)μ 2 + (N−2)μ 2 + λ−λ 1 (Ω)≥0 - frac{left(N-2)mu }{2} + frac{left(N-2)mu }{2}logleft (- frac{left(N-2)mu }{2}right)+ lambda{ - }{}{lambda} ＿1left (Omega) ge 0的不存在性结果。与brsamzis和Nirenberg的研究结果比较(含临界Sobolev指数的非线性椭圆方程的正解)。数学。36(1983)，437-477)，当对数扰动上的参数μ mu不为零时，会出现一些新的有趣现象。
{"title":"The existence of positive solution for an elliptic problem with critical growth and logarithmic perturbation","authors":"Yinbin Deng, Qihan He, Yiqing Pan, X. Zhong","doi":"10.1515/ans-2022-0049","DOIUrl":"https://doi.org/10.1515/ans-2022-0049","url":null,"abstract":"Abstract We consider the existence and nonexistence of the positive solution for the following Brézis-Nirenberg problem with logarithmic perturbation: − Δ u = ∣ u ∣ 2 ∗ − 2 u + λ u + μ u log u 2 x ∈ Ω , u = 0 x ∈ ∂ Ω , left{phantom{rule[-1.25em]{}{0ex}}begin{array}{ll}-Delta u={| u| }^{{2}^{ast }-2}u+lambda u+mu ulog {u}^{2}hspace{1.0em}& xin Omega , u=0hspace{1.0em}& xin partial Omega ,end{array}right. where Ω ⊂ R N Omega subset {{mathbb{R}}}^{N} is a bounded open domain, λ , μ ∈ R lambda ,mu in {mathbb{R}} , N ≥ 3 Nge 3 and 2 ∗ ≔ 2 N N − 2 {2}^{ast }:= frac{2N}{N-2} is the critical Sobolev exponent for the embedding H 0 1 ( Ω ) ↪ L 2 ∗ ( Ω ) {H}_{0}^{1}left(Omega )hspace{0.33em}hookrightarrow hspace{0.33em}{L}^{{2}^{ast }}left(Omega ) . The uncertainty of the sign of s log s 2 slog {s}^{2} in ( 0 , + ∞ ) left(0,+infty ) has some interest in itself. We will show the existence of positive ground state solution, which is of mountain pass type provided λ ∈ R , μ > 0 lambda in {mathbb{R}},mu gt 0 and N ≥ 4 Nge 4 . While the case of μ < 0 mu lt 0 is thornier. However, for N = 3 , 4 N=3,4 , λ ∈ ( − ∞ , λ 1 ( Ω ) ) lambda in left(-infty ,{lambda }_{1}left(Omega )) , we can also establish the existence of positive solution under some further suitable assumptions. A nonexistence result is also obtained for μ < 0 mu lt 0 and − ( N − 2 ) μ 2 + ( N − 2 ) μ 2 log − ( N − 2 ) μ 2 + λ − λ 1 ( Ω ) ≥ 0 -frac{left(N-2)mu }{2}+frac{left(N-2)mu }{2}log left(-frac{left(N-2)mu }{2}right)+lambda -{lambda }_{1}left(Omega )ge 0 if N ≥ 3 Nge 3 . Comparing with the results in the study by Brézis and Nirenberg (Positive solutions of nonlinear elliptic equations involving critical Sobolev exponents, Comm. Pure Appl. Math. 36 (1983), 437–477), some new interesting phenomenon occurs when the parameter μ mu on logarithmic perturbation is not zero.","PeriodicalId":7191,"journal":{"name":"Advanced Nonlinear Studies","volume":" ","pages":""},"PeriodicalIF":1.8,"publicationDate":"2022-10-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44368618","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}
Abstract In this article, we consider the homogeneous complex k k -Hessian equation in an exterior domain C n ⧹ Ω {{mathbb{C}}}^{n}setminus Omega . We prove the existence and uniqueness of the C 1 , 1 {C}^{1,1} solution by constructing approximating solutions. The key point for us is to establish the uniform gradient estimate and the second-order estimate.
{"title":"The exterior Dirichlet problem for the homogeneous complex k-Hessian equation","authors":"Zhenghuan Gao, Xinan Ma, Dekai Zhang","doi":"10.1515/ans-2022-0039","DOIUrl":"https://doi.org/10.1515/ans-2022-0039","url":null,"abstract":"Abstract In this article, we consider the homogeneous complex k k -Hessian equation in an exterior domain C n ⧹ Ω {{mathbb{C}}}^{n}setminus Omega . We prove the existence and uniqueness of the C 1 , 1 {C}^{1,1} solution by constructing approximating solutions. The key point for us is to establish the uniform gradient estimate and the second-order estimate.","PeriodicalId":7191,"journal":{"name":"Advanced Nonlinear Studies","volume":" ","pages":""},"PeriodicalIF":1.8,"publicationDate":"2022-08-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45449270","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}
Abstract In this article, we prove that, generically in the sense of domain variations, any solution to a nonlinear eigenvalue problem is either nondegenerate or the Crandall-Rabinowitz transversality condition that is satisfied. We then deduce that, generically, the unbounded Rabinowitz continuum of solutions is a simple analytic curve. The global bifurcation diagram resembles the classic model case of the Gel’fand problem in two dimensions.
{"title":"Generic properties of the Rabinowitz unbounded continuum","authors":"D. Bartolucci, Yeyao Hu, Aleks Jevnikar, Wen Yang","doi":"10.1515/ans-2022-0062","DOIUrl":"https://doi.org/10.1515/ans-2022-0062","url":null,"abstract":"Abstract In this article, we prove that, generically in the sense of domain variations, any solution to a nonlinear eigenvalue problem is either nondegenerate or the Crandall-Rabinowitz transversality condition that is satisfied. We then deduce that, generically, the unbounded Rabinowitz continuum of solutions is a simple analytic curve. The global bifurcation diagram resembles the classic model case of the Gel’fand problem in two dimensions.","PeriodicalId":7191,"journal":{"name":"Advanced Nonlinear Studies","volume":" ","pages":""},"PeriodicalIF":1.8,"publicationDate":"2022-07-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43582580","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}
Abstract This article introduces a functional generalizing Perelman’s weighted Hilbert-Einstein action and the Dirichlet energy for spinors. It is well defined on a wide class of noncompact manifolds; on asymptotically Euclidean manifolds, the functional is shown to admit a unique critical point, which is necessarily of min-max type, and the Ricci flow is its gradient flow. The proof is based on variational formulas for weighted spinorial functionals, valid on all spin manifolds with boundary.
{"title":"The spinorial energy for asymptotically Euclidean Ricci flow","authors":"Julius Baldauf, Tristan Ozuch","doi":"10.1515/ans-2022-0045","DOIUrl":"https://doi.org/10.1515/ans-2022-0045","url":null,"abstract":"Abstract This article introduces a functional generalizing Perelman’s weighted Hilbert-Einstein action and the Dirichlet energy for spinors. It is well defined on a wide class of noncompact manifolds; on asymptotically Euclidean manifolds, the functional is shown to admit a unique critical point, which is necessarily of min-max type, and the Ricci flow is its gradient flow. The proof is based on variational formulas for weighted spinorial functionals, valid on all spin manifolds with boundary.","PeriodicalId":7191,"journal":{"name":"Advanced Nonlinear Studies","volume":" ","pages":""},"PeriodicalIF":1.8,"publicationDate":"2022-06-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46531868","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}
Abstract The goal of this article is to give a new proof of the wave trace formula proved by Richard Melrose in an impressive article. This trace formula is an extension of the Chazarain-Duistermaat-Guillemin trace formula (denoted as “CDG trace formula” in this article) to the case of a sub-Riemannian Laplacian on a 3D contact closed manifold. The proof uses a normal form constructed in previous papers, following the pioneering work of Melrose to reduce the case of the invariant Laplacian on the 3D-Heisenberg group. We need also the propagation of singularities results of the works of Ivrii, Lascar, and Melrose.
{"title":"A proof of a trace formula by Richard Melrose","authors":"Yves Colin de Verdière","doi":"10.1515/ans-2022-0054","DOIUrl":"https://doi.org/10.1515/ans-2022-0054","url":null,"abstract":"Abstract The goal of this article is to give a new proof of the wave trace formula proved by Richard Melrose in an impressive article. This trace formula is an extension of the Chazarain-Duistermaat-Guillemin trace formula (denoted as “CDG trace formula” in this article) to the case of a sub-Riemannian Laplacian on a 3D contact closed manifold. The proof uses a normal form constructed in previous papers, following the pioneering work of Melrose to reduce the case of the invariant Laplacian on the 3D-Heisenberg group. We need also the propagation of singularities results of the works of Ivrii, Lascar, and Melrose.","PeriodicalId":7191,"journal":{"name":"Advanced Nonlinear Studies","volume":" ","pages":""},"PeriodicalIF":1.8,"publicationDate":"2022-05-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48309193","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}
Abstract We obtain a comparison formula for integrals of mean curvatures of Riemannian hypersurfaces via Reilly’s identities. As applications, we derive several geometric inequalities for a convex hypersurface Γ Gamma in a Cartan-Hadamard manifold M M . In particular, we show that the first mean curvature integral of a convex hypersurface γ gamma nested inside Γ Gamma cannot exceed that of Γ Gamma , which leads to a sharp lower bound for the total first mean curvature of Γ Gamma in terms of the volume it bounds in M M in dimension 3. This monotonicity property is extended to all mean curvature integrals when γ gamma is parallel to Γ Gamma , or M M has constant curvature. We also characterize hyperbolic balls as minimizers of the mean curvature integrals among balls with equal radii in Cartan-Hadamard manifolds.
{"title":"Total mean curvatures of Riemannian hypersurfaces","authors":"M. Ghomi, J. Spruck","doi":"10.1515/ans-2022-0029","DOIUrl":"https://doi.org/10.1515/ans-2022-0029","url":null,"abstract":"Abstract We obtain a comparison formula for integrals of mean curvatures of Riemannian hypersurfaces via Reilly’s identities. As applications, we derive several geometric inequalities for a convex hypersurface Γ Gamma in a Cartan-Hadamard manifold M M . In particular, we show that the first mean curvature integral of a convex hypersurface γ gamma nested inside Γ Gamma cannot exceed that of Γ Gamma , which leads to a sharp lower bound for the total first mean curvature of Γ Gamma in terms of the volume it bounds in M M in dimension 3. This monotonicity property is extended to all mean curvature integrals when γ gamma is parallel to Γ Gamma , or M M has constant curvature. We also characterize hyperbolic balls as minimizers of the mean curvature integrals among balls with equal radii in Cartan-Hadamard manifolds.","PeriodicalId":7191,"journal":{"name":"Advanced Nonlinear Studies","volume":" ","pages":""},"PeriodicalIF":1.8,"publicationDate":"2022-04-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42368893","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}
Abstract In this article, we revisit previous Pogorelov-type interior and global second derivative estimates of N. S. Trudinger, F. Jiang, and J. Liu for solutions of Monge-Ampère-type partial differential equations. Taking account of recent strict convexity regularity results of Guillen-Kitagawa and Rankin and following our earlier work in the optimal transportation case, we remove the monotonicity assumptions in the more general case of generated Jacobian equations and consequently in the subsequent application to classical solvability and global regularity for second boundary value problems.
{"title":"A note on second derivative estimates for Monge-Ampère-type equations","authors":"N. Trudinger","doi":"10.1515/ans-2022-0036","DOIUrl":"https://doi.org/10.1515/ans-2022-0036","url":null,"abstract":"Abstract In this article, we revisit previous Pogorelov-type interior and global second derivative estimates of N. S. Trudinger, F. Jiang, and J. Liu for solutions of Monge-Ampère-type partial differential equations. Taking account of recent strict convexity regularity results of Guillen-Kitagawa and Rankin and following our earlier work in the optimal transportation case, we remove the monotonicity assumptions in the more general case of generated Jacobian equations and consequently in the subsequent application to classical solvability and global regularity for second boundary value problems.","PeriodicalId":7191,"journal":{"name":"Advanced Nonlinear Studies","volume":" ","pages":""},"PeriodicalIF":1.8,"publicationDate":"2022-04-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47872421","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}
Abstract We observe that the k k -dimensional width of an n n -ball in a space form is given by the area of an equatorial k k -ball. We also discuss the relationship between widths and lower bounds for the area of a free boundary minimal submanifold in a space form ball.
{"title":"Widths of balls and free boundary minimal submanifolds","authors":"Jonathan J. Zhu","doi":"10.1515/ans-2022-0044","DOIUrl":"https://doi.org/10.1515/ans-2022-0044","url":null,"abstract":"Abstract We observe that the k k -dimensional width of an n n -ball in a space form is given by the area of an equatorial k k -ball. We also discuss the relationship between widths and lower bounds for the area of a free boundary minimal submanifold in a space form ball.","PeriodicalId":7191,"journal":{"name":"Advanced Nonlinear Studies","volume":" ","pages":""},"PeriodicalIF":1.8,"publicationDate":"2022-03-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46199573","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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