The Experts below are selected from a list of 47748 Experts worldwide ranked by ideXlab platform
Blaine H Lawson - One of the best experts on this subject based on the ideXlab platform.
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Dirichlet duality and the nonlinear Dirichlet Problem on riemannian manifolds
arXiv: Analysis of PDEs, 2009Co-Authors: Reese F Harvey, Blaine H LawsonAbstract:In this paper we study the Dirichlet Problem for fully nonlinear second-order equations on a riemannian manifold. As in a previous paper we define equations via closed subsets of the 2-jet bundle. Basic existence and uniqueness theorems are established in a wide variety of settings. However, the emphasis is on starting with a constant coefficient equation as a model, which then universally determines an equation on every riemannian manifold which is equipped with a topological reduction of the structure group to the invariance group of the model. For example, this covers all branches of the homogeneous complex Monge-Ampere equation on an almost complex hermitian manifold X. In general, for an equation F on a manifold X and a smooth domain D in X, we give geometric conditions which imply that the Dirichlet Problem on D is uniquely solvable for all continuous boundary functions. We begin by introducing a weakened form of comparison which has the advantage that local implies global. We then associate to F two natural "conical subequations": a monotonicity subequation M and the asymptotic interior of F. If X carries a global M-subharmonic function, then weak comparison implies full comparison. The asymptotic interior of F is used to formulate boundary convexity and provide barriers. In combination the Dirichlet Problem becomes uniquely solvable as claimed. A considerable portion of the paper is concerned with specific examples. They include a wide variety of equations which make sense on any riemannian manifold, and many which hold universally on almost complex or quaternionic hermitian manifolds, or topologically calibrated manifolds.
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Dirichlet duality and the nonlinear Dirichlet Problem
Communications on Pure and Applied Mathematics, 2009Co-Authors: Reese F Harvey, Blaine H LawsonAbstract:We study the Dirichlet Problem for fully nonlinear, degenerate elliptic equations of the form F(Hess u) = 0 on a smoothly bounded domain Ω ⋐ ℝn. In our approach the equation is replaced by a subset F ⊂ Sym2(ℝn) of the symmetric n × n matrices with ∂F ⊆ {F = 0}. We establish the existence and uniqueness of continuous solutions under an explicit geometric “F-convexity” assumption on the boundary ∂Ω. We also study the topological structure of F-convex domains and prove a theorem of Andreotti-Frankel type. Two key ingredients in the analysis are the use of “subaffine functions” and “Dirichlet duality.” Associated to F is a Dirichlet dual set F that gives a dual Dirichlet Problem. This pairing is a true duality in that the dual of F is F, and in the analysis the roles of F and F are interchangeable. The duality also clarifies many features of the Problem including the appropriate conditions on the boundary. Many interesting examples are covered by these results including: all branches of the homogeneous Monge-Ampere equation over ℝ, ℂ, and ℍ; equations appearing naturally in calibrated geometry, Lagrangian geometry, and p-convex Riemannian geometry; and all branches of the special Lagrangian potential equation. © 2008 Wiley Periodicals, Inc.
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Dirichlet duality and the nonlinear Dirichlet Problem
arXiv: Analysis of PDEs, 2007Co-Authors: Reese F Harvey, Blaine H LawsonAbstract:We study the Dirichlet Problem for fully nonlinear, degenerate elliptic equations of the form f(Hess, u)=0 on a smoothly bounded domain D in R^n. In our approach the equation is replaced by a subset F of the space of symmetric nxn-matrices, with bdy(F) contined in the set {f=0}. We establish the existence and uniqueness of continuous solutions under an explicit geometric ``F-convexity'' assumption on the boundary bdy(F). The topological structure of F-convex domains is also studied and a theorem of Andreotti-Frankel type is proved for them. Two key ingredients in the analysis are the use of subaffine functions and Dirichlet duality, both introduced here. Associated to F is a Dirichlet dual set F* which gives a dual Dirichlet Problem. This pairing is a true duality in that the dual of F* is F and in the analysis the roles of F and F* are interchangeable. The duality also clarifies many features of the Problem including the appropriate conditions on the boundary. Many interesting examples are covered by these results including: All branches of the homogeneous Monge-Ampere equation over R, C and H; equations appearing naturally in calibrated geometry, Lagrangian geometry and p-convex riemannian geometry, and all branches of the Special Lagrangian potential equation.
Lei Qiao - One of the best experts on this subject based on the ideXlab platform.
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The Dirichlet Problem in a generalized strip
SCIENTIA SINICA Mathematica, 2013Co-Authors: Lei Qiao, Guantie DengAbstract:In this paper, we give the integral representation for solutions to Dirichlet Problem in a generalized strip. Meanwhile, we give a necessary and sufficient condition for a class of functions which is harmonic in this strip and vanishes on its boundary.
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The Dirichlet Problem on the Upper Half-Space
Abstract and Applied Analysis, 2012Co-Authors: Jinjin Huang, Lei QiaoAbstract:A solution of the Dirichlet Problem on the upper half-space is constructed by the generalized Dirichlet integral with a fast-growing continuous boundary function.
Reese F Harvey - One of the best experts on this subject based on the ideXlab platform.
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Dirichlet duality and the nonlinear Dirichlet Problem on riemannian manifolds
arXiv: Analysis of PDEs, 2009Co-Authors: Reese F Harvey, Blaine H LawsonAbstract:In this paper we study the Dirichlet Problem for fully nonlinear second-order equations on a riemannian manifold. As in a previous paper we define equations via closed subsets of the 2-jet bundle. Basic existence and uniqueness theorems are established in a wide variety of settings. However, the emphasis is on starting with a constant coefficient equation as a model, which then universally determines an equation on every riemannian manifold which is equipped with a topological reduction of the structure group to the invariance group of the model. For example, this covers all branches of the homogeneous complex Monge-Ampere equation on an almost complex hermitian manifold X. In general, for an equation F on a manifold X and a smooth domain D in X, we give geometric conditions which imply that the Dirichlet Problem on D is uniquely solvable for all continuous boundary functions. We begin by introducing a weakened form of comparison which has the advantage that local implies global. We then associate to F two natural "conical subequations": a monotonicity subequation M and the asymptotic interior of F. If X carries a global M-subharmonic function, then weak comparison implies full comparison. The asymptotic interior of F is used to formulate boundary convexity and provide barriers. In combination the Dirichlet Problem becomes uniquely solvable as claimed. A considerable portion of the paper is concerned with specific examples. They include a wide variety of equations which make sense on any riemannian manifold, and many which hold universally on almost complex or quaternionic hermitian manifolds, or topologically calibrated manifolds.
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Dirichlet duality and the nonlinear Dirichlet Problem
Communications on Pure and Applied Mathematics, 2009Co-Authors: Reese F Harvey, Blaine H LawsonAbstract:We study the Dirichlet Problem for fully nonlinear, degenerate elliptic equations of the form F(Hess u) = 0 on a smoothly bounded domain Ω ⋐ ℝn. In our approach the equation is replaced by a subset F ⊂ Sym2(ℝn) of the symmetric n × n matrices with ∂F ⊆ {F = 0}. We establish the existence and uniqueness of continuous solutions under an explicit geometric “F-convexity” assumption on the boundary ∂Ω. We also study the topological structure of F-convex domains and prove a theorem of Andreotti-Frankel type. Two key ingredients in the analysis are the use of “subaffine functions” and “Dirichlet duality.” Associated to F is a Dirichlet dual set F that gives a dual Dirichlet Problem. This pairing is a true duality in that the dual of F is F, and in the analysis the roles of F and F are interchangeable. The duality also clarifies many features of the Problem including the appropriate conditions on the boundary. Many interesting examples are covered by these results including: all branches of the homogeneous Monge-Ampere equation over ℝ, ℂ, and ℍ; equations appearing naturally in calibrated geometry, Lagrangian geometry, and p-convex Riemannian geometry; and all branches of the special Lagrangian potential equation. © 2008 Wiley Periodicals, Inc.
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Dirichlet duality and the nonlinear Dirichlet Problem
arXiv: Analysis of PDEs, 2007Co-Authors: Reese F Harvey, Blaine H LawsonAbstract:We study the Dirichlet Problem for fully nonlinear, degenerate elliptic equations of the form f(Hess, u)=0 on a smoothly bounded domain D in R^n. In our approach the equation is replaced by a subset F of the space of symmetric nxn-matrices, with bdy(F) contined in the set {f=0}. We establish the existence and uniqueness of continuous solutions under an explicit geometric ``F-convexity'' assumption on the boundary bdy(F). The topological structure of F-convex domains is also studied and a theorem of Andreotti-Frankel type is proved for them. Two key ingredients in the analysis are the use of subaffine functions and Dirichlet duality, both introduced here. Associated to F is a Dirichlet dual set F* which gives a dual Dirichlet Problem. This pairing is a true duality in that the dual of F* is F and in the analysis the roles of F and F* are interchangeable. The duality also clarifies many features of the Problem including the appropriate conditions on the boundary. Many interesting examples are covered by these results including: All branches of the homogeneous Monge-Ampere equation over R, C and H; equations appearing naturally in calibrated geometry, Lagrangian geometry and p-convex riemannian geometry, and all branches of the Special Lagrangian potential equation.
Jon Wilkening - One of the best experts on this subject based on the ideXlab platform.
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eigenfunctions and the Dirichlet Problem for the classical kimura diffusion operator
Siam Journal on Applied Mathematics, 2017Co-Authors: Charles L Epstein, Jon WilkeningAbstract:We study the classical Kimura diffusion operator defined on the $n$-simplex, $\operatorname{L^{Kim}}=\sum_{1\leq i,j\leq n+1}x_i(\delta_{ij}-x_j)\partial_{x_i}\partial_{x_j},$ which has important applications in population genetics. Because it is a degenerate elliptic operator acting on a singular space, special tools are required to analyze and construct solutions to elliptic and parabolic Problems defined by this operator. The natural boundary value Problems are the “regular” Problem and the Dirichlet Problem. For the regular Problem, one can only specify the regularity of the solution at the boundary. For the Dirichlet Problem, one can specify the boundary values, but the solution is then not smooth at the boundary. In this paper we give a computationally effective recursive method to construct the eigenfunctions of the regular operator in any dimension, and a recursive method to use them to solve the inhomogeneous equation. As noted, the Dirichlet Problem does not have a regular solution. We give an e...
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eigenfunctions and the Dirichlet Problem for the classical kimura diffusion operator
arXiv: Analysis of PDEs, 2015Co-Authors: Charles L Epstein, Jon WilkeningAbstract:We study the classical Kimura diffusion operator defined on the n-simplex, $$L^{Kim}=\sum_{1\leq i,j\leq n+1}x_ix_j\partial_{x_i}\partial_{x_j}$$ We give novel constructions for the basis of eigenpolynomials, and the solution to the inhomogeneous Dirichlet Problem, which are well adapted to numerical applications. Our solution of the Dirichlet Problem is quite explicit and provides a precise description of the singularities that arise along the boundary.
Jinjin Huang - One of the best experts on this subject based on the ideXlab platform.
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The Dirichlet Problem on the Upper Half-Space
Abstract and Applied Analysis, 2012Co-Authors: Jinjin Huang, Lei QiaoAbstract:A solution of the Dirichlet Problem on the upper half-space is constructed by the generalized Dirichlet integral with a fast-growing continuous boundary function.
