scholarly journals Extremality Conditions and Regularity of Solutions to Optimal Partition Problems Involving Laplacian Eigenvalues

2015 ◽  
Vol 220 (1) ◽  
pp. 363-443 ◽  
Author(s):  
Miguel Ramos ◽  
Hugo Tavares ◽  
Susanna Terracini
Keyword(s):  
Regularity Of Solutions ◽  
Optimal Partition ◽  
Laplacian Eigenvalues ◽  
Extremality Conditions

Limit Configurations of Schrödinger Systems Versus Optimal Partition for the Principal Eigenvalue of Elliptic Systems

2019 ◽  
Vol 19 (4) ◽  
pp. 693-715 ◽  
Author(s):  
Haijun Luo ◽  
Zhitao Zhang
Keyword(s):  
Principal Eigenvalue ◽  
Elliptic Systems ◽  
Coupling Constants ◽  
Optimal Partition ◽  
Minimizing Sequences ◽  
Linear Coupling ◽  
Existence Of Positive Solutions ◽  
Schrödinger Systems ◽  
Extremality Conditions ◽  
Schrödinger System

AbstractWe study a Schrödinger system of four equations with linear coupling functions and nonlinear couplings, including the case that the corresponding elliptic operators are indefinite. For any given nonlinear coupling {\beta>0}, we first use minimizing sequences on a normalized set to obtain a minimizer, which implies the existence of positive solutions for some linear coupling constants {\mu_{\beta},\nu_{\beta}} by Lagrange multiplier rules. Then, as {\beta\to\infty}, we prove that the limit configurations to the competing system are segregated in two groups, develop a variant of Almgren’s monotonicity formula to reveal the Lipschitz continuity of the limit profiles and establish a kind of local Pohozaev identity to obtain the extremality conditions. Finally, we study the relation between the limit profiles and the optimal partition for principal eigenvalue of the elliptic system and obtain an optimal partition for principal eigenvalues of elliptic systems.

scholarly journals Existence results for nonlinear degenerate elliptic equations with lower order terms

2020 ◽  
Vol 10 (1) ◽  
pp. 301-310
Author(s):  
Weilin Zou ◽  
Xinxin Li
Keyword(s):  
Lower Order ◽  
Elliptic Equations ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Regularity Of Solutions ◽  
Degenerate Elliptic Equations ◽  
Degenerate Elliptic ◽  
Initial Boundary Value ◽  
Lower Order Terms ◽  
Nonlinear Degenerate

Abstract In this paper, we prove the existence and regularity of solutions of the homogeneous Dirichlet initial-boundary value problem for a class of degenerate elliptic equations with lower order terms. The results we obtained here, extend some existing ones of [2, 9, 11] in some sense.

Boundedness of multi-parameter pseudo-differential operators on multi-parameter local Hardy spaces

2020 ◽  
Vol 32 (4) ◽  
pp. 919-936 ◽  
Author(s):  
Jiao Chen ◽  
Wei Ding ◽  
Guozhen Lu
Keyword(s):  
Hardy Spaces ◽  
Differential Operators ◽  
Integral Operators ◽  
Fourier Multipliers ◽  
Regularity Of Solutions ◽  
Local Version ◽  
Fourier Integral Operators ◽  
Pseudo Differential Operators ◽  
Local Hardy Spaces ◽  
The One

AbstractAfter the celebrated work of L. Hörmander on the one-parameter pseudo-differential operators, the applications of pseudo-differential operators have played an important role in partial differential equations, geometric analysis, harmonic analysis, theory of several complex variables and other branches of modern analysis. For instance, they are used to construct parametrices and establish the regularity of solutions to PDEs such as the {\overline{\partial}} problem. The study of Fourier multipliers, pseudo-differential operators and Fourier integral operators has stimulated further such applications. It is well known that the one-parameter pseudo-differential operators are {L^{p}({\mathbb{R}^{n}})} bounded for {1<p<\infty}, but only bounded on local Hardy spaces {h^{p}({\mathbb{R}^{n}})} introduced by Goldberg in [D. Goldberg, A local version of real Hardy spaces, Duke Math. J. 46 1979, 1, 27–42] for {0<p\leq 1}. Though much work has been done on the {L^{p}(\mathbb{R}^{n_{1}}\times\mathbb{R}^{n_{2}})} boundedness for {1<p<\infty} and Hardy {H^{p}(\mathbb{R}^{n_{1}}\times\mathbb{R}^{n_{2}})} boundedness for {0<p\leq 1} for multi-parameter Fourier multipliers and singular integral operators, not much has been done yet for the boundedness of multi-parameter pseudo-differential operators in the range of {0<p\leq 1}. The main purpose of this paper is to establish the boundedness of multi-parameter pseudo-differential operators on multi-parameter local Hardy spaces {h^{p}(\mathbb{R}^{n_{1}}\times\mathbb{R}^{n_{2}})} for {0<p\leq 1} recently introduced by Ding, Lu and Zhu in [W. Ding, G. Lu and Y. Zhu, Multi-parameter local Hardy spaces, Nonlinear Anal. 184 2019, 352–380].

scholarly journals The Dirichlet problem for the Jacobian equation in critical and supercritical Sobolev spaces

2021 ◽  
Vol 60 (1) ◽  
Author(s):  
André Guerra ◽  
Lukas Koch ◽  
Sauli Lindberg
Keyword(s):  
Dirichlet Problem ◽  
Sobolev Spaces ◽  
Regularity Of Solutions ◽  
Jacobian Equation ◽  
L Log L

AbstractWe study existence and regularity of solutions to the Dirichlet problem for the prescribed Jacobian equation, $$\det D u =f$$ det D u = f , where f is integrable and bounded away from zero. In particular, we take $$f\in L^p$$ f ∈ L p , where $$p>1$$ p > 1 , or in $$L\log L$$ L log L . We prove that for a Baire-generic f in either space there are no solutions with the expected regularity.

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