Existence of distributional solutions of a closed Dirichlet problem for an elliptic–hyperbolic equation

2011 ◽  
Vol 74 (17) ◽  
pp. 6512-6517 ◽  
Author(s):  
Meng Xu ◽  
Xiaoping Yang
Keyword(s):  
Dirichlet Problem ◽  
Hyperbolic Equation ◽  
Distributional Solutions

On the Correctness of the Dirichlet Problem in a Characteristic Rectangle for Fourth Order Linear Hyperbolic Equations

1999 ◽  
Vol 6 (5) ◽  
pp. 447-470
Author(s):  
T. Kiguradze
Keyword(s):  
Dirichlet Problem ◽  
Hyperbolic Equation ◽  
Fourth Order ◽  
Trivial Solution ◽  
Hyperbolic Equations ◽  
Homogeneous Problem ◽  
Linear Hyperbolic Equation ◽  
Order Linear ◽  
Linear Hyperbolic Equations

Abstract It is proved that the Dirichlet problem is correct in the characteristic rectangle 𝐷𝑎𝑏 = [0, 𝑎] × [0, 𝑏] for the linear hyperbolic equation with the summable in 𝐷𝑎𝑏 coefficients 𝑝0, 𝑝1, 𝑝2, 𝑝3 and 𝑞 if and only if the corresponding homogeneous problem has only the trivial solution. The effective and optimal in some sense restrictions on 𝑝0, 𝑝1, 𝑝2 and 𝑝3 guaranteeing the correctness of the Dirichlet problem are established.

Note on a maximum principle and a uniqueness theorem for an elliptic-hyperbolic equation

1956 ◽  
Vol 236 (1204) ◽  
pp. 141-144 ◽  
Keyword(s):  
Dirichlet Problem ◽  
Maximum Principle ◽  
Elliptic Equation ◽  
Hyperbolic Equation ◽  
Uniqueness Theorem ◽  
Mixed Type ◽  
Boundary Curve ◽  
Type K ◽  
Equation Of Mixed Type

A maximum principle is proved for the function ψ = J [ — 2u x u y dx + (Ku 2 x — u 2 y ) dy], where u is a solution of the equation of mixed type K(y)u xx + u yy = 0 with K(y) ≷ 0 for y ≷ 0. The proof rests in showing that iff satisfies an elliptic equation for y > 0 and that it is a non-decreasing function of y for y ⩽ 0. This maximum principle leads to a uniqueness theorem for the appropriate analogue to the Dirichlet problem for mixed equations under some conditions on the shape of the boundary curve. Very weak restrictions are imposed on K(y).

Dirichlet, Neumann and mixed Dirichlet-Neumann boundary value problems for Uxy = 0 in rectangles

1978 ◽  
Vol 82 (1-2) ◽  
pp. 107-110 ◽  
Author(s):  
Ali I. Abdul-Latif
Keyword(s):  
Dirichlet Problem ◽  
Hyperbolic Equation ◽  
Boundary Value Problems ◽  
Hyperbolic Equations ◽  
Boundary Value ◽  
Neumann Boundary ◽  
Partial Answer ◽  
Neumann Boundary Value Problems ◽  
Well Posed

SynopsisIt is well known that the Dirichlet problem for hyperbolic equations is a classical “not well posed” problem. Here we consider the Dirichlet, Neumann and mixed Dirichlet-Neumann boundary value problems for the hyperbolic equation uxy = 0 in all positions of the square and a class of rectangles. We also get a partial answer to the problem which deals with a ray that moves from any point on the boundary of a rectangle and is reflected on the boundary such that the angle between every ray and its reflection is π/2.

Dirichlet problem for semilinear hyperbolic equation

2005 ◽  
Vol 63 (5-7) ◽  
pp. e43-e52 ◽  
Author(s):  
Andrzej Nowakowski ◽  
Iwona Nowakowska
Keyword(s):  
Dirichlet Problem ◽  
Hyperbolic Equation ◽  
Semilinear Hyperbolic Equation

scholarly journals Remarks on the Nonlocal Dirichlet Problem

2020 ◽  
Vol 54 (1) ◽  
pp. 119-151
Author(s):  
Tomasz Grzywny ◽  
Moritz Kassmann ◽  
Łukasz Leżaj
Keyword(s):  
Dirichlet Problem ◽  
Lévy Processes ◽  
Sufficient Conditions ◽  
Representation Formula ◽  
Levy Processes ◽  
Classical Representation ◽  
Translation Invariant ◽  
Classical Sense ◽  
Distributional Solutions ◽  
Integrodifferential Operators

AbstractWe study translation-invariant integrodifferential operators that generate Lévy processes. First, we investigate different notions of what a solution to a nonlocal Dirichlet problem is and we provide the classical representation formula for distributional solutions. Second, we study the question under which assumptions distributional solutions are twice differentiable in the classical sense. Sufficient conditions and counterexamples are provided.

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