Behavior of solutions of the Dirichlet Problem for the $p(x)$-Laplacian at a boundary point

2020 ◽  
Vol 31 (2) ◽  
pp. 251-271 ◽  
Author(s):  
Yu. A. Alkhutov ◽  
M. D. Surnachev
Keyword(s):  
Dirichlet Problem ◽  
Boundary Point

scholarly journals On Exceptional Values of a Meromorphic Function

1955 ◽  
Vol 9 ◽  
pp. 119-121
Author(s):  
Makoto Ohtsuka
Keyword(s):  
Dirichlet Problem ◽  
Meromorphic Function ◽  
Harmonic Measure ◽  
Boundary Point ◽  
Measure Zero ◽  
Regular Point ◽  
Open Set

M. Brelot [1] has shown that if u(z) is subharmonic in an open set D in the z-plane with boundary C and is bounded from above in a neighborhood of a boundary point z0, which is contained in a set E ⊂ C of inner harmonic measure zero with respect to D, and such that z0 is a regular point for Dirichlet problem in D, then(1) .

Estimates for solutions of the Dirichlet problem for the biharmonic equation in a neighbourhood of an irregular boundary point and in a neighbourhood of infinity. Saint-Venant's principle

1983 ◽  
Vol 93 (3-4) ◽  
pp. 327-343 ◽  
Author(s):  
Y. A. Kondratiev ◽  
O. A. Oleinik
Keyword(s):  
Dirichlet Problem ◽  
Boundary Conditions ◽  
Boundary Point ◽  
Biharmonic Equation ◽  
Maximum Modulus ◽  
Integral Inequalities ◽  
Energy Estimates ◽  
Irregular Boundary ◽  
Continuity Of Solutions

SynopsisIn this paper energy estimates for solutions of the Dirichlet problem for the biharmonicequation, expressing Saint-Venant's principle in elasticity, are proved. From these integral inequalities, estimates for the maximum modulus of solutions and the gradient of solutions with homogeneous Diriehlet's boundary conditions in a neighbourhood of an irregular boundary point or in a neighbourhood of infinity are derived. These estimates characterize the continuity of solutions and their gradients at these points.

Solvability of one nonlocal Dirichlet problem for the Poisson equation

2019 ◽  
Vol 50 (1) ◽  
pp. 67-88
Author(s):  
Batirkhan Turmetov ◽  
Valery Karachik
Keyword(s):  
Dirichlet Problem ◽  
Poisson Equation ◽  
The Poisson Equation
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