In this paper, we prove global second derivative estimates for solutions of the Dirichlet problem for the Monge-Amp`re equation when the inhomogee neous term is only assumed to be H¨lder continuous. As a consequence of our o approach, we also establish the existence and uniqueness of globally smooth solutions to the second boundary value problem for the aﬃne maximal surface equation and aﬃne mean curvature equation.
MULTIDIMENSIONAL KOLMOGOROV-PETROVSKY TEST FOR THE BOUNDARY REGULARITY AND IRREGULARITY OF SOLUTIONS TO THE HEAT EQUATION
UGUR G. ABDULLA Received 25 August 2004 Dedicated to I. G. Petrovsky
This paper establishes necessary and suﬃcient condition for the regularity of a characteristic top boundary point of an arbitrary open subset of RN+1 (N ≥ 2) for the diﬀusion (or heat) equation. The result implies asymptotic probability law for the standard Ndimensional Brownian motion. 1. Introduction and main result Consider the domain Ωδ = (x,t) ∈ RN+1 : |x| 0, N ≥ 2, x = (x1 ,...
Annals of Mathematics
We study the motion of an incompressible perfect liquid body in vacuum. This can be thought of as a model for the motion of the ocean or a star. The free surface moves with the velocity of the liquid and the pressure vanishes on the free surface. This leads to a free boundary problem for Euler’s equations, where the regularity of the boundary enters to highest order. We prove local existence in Sobolev spaces assuming a “physical condition”, related to the fact that the pressure of a ﬂuid has to be positive. ...
In this paper we study Lipschitz solutions of partial diﬀerential relations of the form (1) ∇u(x) ∈ K a.e. in Ω,
where u is a (Lipschitz) mapping of an open set Ω ⊂ Rn into Rm , ∇u(x) is its gradient (i.e. the matrix ∂ui (x)/∂xj , 1 ≤ i ≤ m, 1 ≤ j ≤ n, deﬁned for almost every x ∈ Ω), and K is a subset of the set M m×n of all real m × n matrices. In addition to relation (1), boundary conditions and other conditions on u will also be considered. Relation (1)...
ON WEAK SOLUTIONS OF THE EQUATIONS OF MOTION OF A VISCOELASTIC MEDIUM WITH VARIABLE BOUNDARY
V. G. ZVYAGIN AND V. P. ORLOV Received 2 September 2005
The regularized system of equations for one model of a viscoelastic medium with memory along trajectories of the ﬁeld of velocities is under consideration. The case of a changing domain is studied. We investigate the weak solvability of an initial boundary value problem for this system. 1. Introduction The purpose of the present paper is an extension of the result of  on the case of a changing domain. Let Ωt ∈ Rn , 2 ≤...
Luong and Loi Boundary Value Problems 2011, 2011:56 http://www.boundaryvalueproblems.com/content/2011/1/56
Initial boundary value problems for second order parabolic systems in cylinders with polyhedral base
Vu Trong Luong1* and Do Van Loi2
* Correspondence: luongvt2003@yahoo.
Tuyển tập báo cáo các nghiên cứu khoa học quốc tế ngành hóa học dành cho các bạn yêu hóa học tham khảo đề tài: Research Article Regularity of Parabolic Hemivariational Inequalities with Boundary Conditions
BOUNDARY VALUE PROBLEMS FOR THE 2ND-ORDER SEIBERG-WITTEN EQUATIONS
CELSO MELCHIADES DORIA Received 8 June 2004
It is shown that the nonhomogeneous Dirichlet and Neuman problems for the 2nd-order Seiberg-Witten equation on a compact 4-manifold X admit a regular solution once the nonhomogeneous Palais-Smale condition is satisﬁed. The approach consists in applying the elliptic techniques to the variational setting of the Seiberg-Witten equation. The gauge invariance of the functional allows to restrict the problem to the Coulomb subspace C of conﬁguration space.
MAXIMAL REGULAR BOUNDARY VALUE PROBLEMS IN BANACH-VALUED WEIGHTED SPACE
RAVI P. AGARWAL, MARTIN BOHNER, AND VELI B. SHAKHMUROV Received 10 July 2004
This study focuses on nonlocal boundary value problems for elliptic ordinary and partial diﬀerential-operator equations of arbitrary order, deﬁned in Banach-valued function spaces. The region considered here has a varying bound and depends on a certain parameter.
Tuyển tập báo cáo các nghiên cứu khoa học quốc tế ngành hóa học dành cho các bạn yêu hóa học tham khảo đề tài: On the regularity of the solution for the second initial boundary value problem for hyperbolic systems in domains with conical points
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