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Convex Domains (convex + domain)

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Selected Abstracts

On blow-up rate for sign-changing solutions in a convex domain

MATHEMATICAL METHODS IN THE APPLIED SCIENCES, Issue 15 2004
Yoshikazu Giga
Abstract This paper studies a growth rate of a solution blowing up at time T of the semilinear heat equation ut , ,u , ,u,p,1u=0 in a convex domain D in ,n with zero-boundary condition. For a subcritical p , (1,(n+2)/(n,2)) a growth rate estimate ,u(x,t),,C(T,t),1/(p,1), x , D, t , (0,T) is established with C independent of t provided that D is uniformly C2. The estimate applies to sign-changing solutions. The same estimate has been recently established when D=,n by authors. The proof is similar but we need to establish Lh , Lk estimate for a time-dependent domain because of the presence of the boundary. Copyright © 2004 John Wiley & Sons, Ltd. [source]

A deterministic-control-based approach motion by curvature,

COMMUNICATIONS ON PURE & APPLIED MATHEMATICS, Issue 3 2006
Robert Kohn
The level-set formulation of motion by mean curvature is a degenerate parabolic equation. We show that its solution can be interpreted as the value function of a deterministic two-person game. More precisely, we give a family of discrete-time, two-person games whose value functions converge in the continuous-time limit to the solution of the motion-by-curvature PDE. For a convex domain, the boundary's "first arrival time" solves a degenerate elliptic equation; this corresponds, in our game-theoretic setting, to a minimum-exit-time problem. For a nonconvex domain the two-person game still makes sense; we draw a connection between its minimum exit time and the evolution of curves with velocity equal to the "positive part of the curvature." These results are unexpected, because the value function of a deterministic control problem is normally the solution of a first-order Hamilton-Jacobi equation. Our situation is different because the usual first-order calculation is singular. © 2005 Wiley Periodicals, Inc. [source]

Dirichlet duality and the nonlinear Dirichlet problem

COMMUNICATIONS ON PURE & APPLIED MATHEMATICS, Issue 3 2009
F. Reese Harvey
We study the Dirichlet problem for fully nonlinear, degenerate elliptic equations of the form F(Hess u) = 0 on a smoothly bounded domain , , ,n. In our approach the equation is replaced by a subset F , Sym2(,n) 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 ,,. 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-Ampère equation over ,, ,, and ,; 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. [source]

The Dirichlet problem for the minimal surface system in arbitrary dimensions and codimensions

COMMUNICATIONS ON PURE & APPLIED MATHEMATICS, Issue 2 2004
Mu-Tao Wang
Let , be a bounded C2 domain in ,n and , ,, , ,m be a continuous map. The Dirichlet problem for the minimal surface system asks whether there exists a Lipschitz map f : , , ,m with f|,, = , and with the graph of f a minimal submanifold in ,n+m. For m = 1, the Dirichlet problem was solved more than 30 years ago by Jenkins and Serrin [12] for any mean convex domains and the solutions are all smooth. This paper considers the Dirichlet problem for convex domains in arbitrary codimension m. We prove that if , : ¯, , ,m satisfies 8n, sup, |D2,| + ,2 sup,, |D,| < 1, then the Dirichlet problem for ,|,, is solvable in smooth maps. Here , is the diameter of ,. Such a condition is necessary in view of an example of Lawson and Osserman [13]. In order to prove this result, we study the associated parabolic system and solve the Cauchy-Dirichlet problem with , as initial data. © 2003 Wiley Periodicals, Inc. [source]