On cauchy's problem for hyperbolic equations and the differentiability of solutions of elliptic equations

1955 ◽  
Vol 8 (4) ◽  
pp. 615-633 ◽  
Author(s):  
Peter D. Lax
Keyword(s):  
Elliptic Equations ◽  
Hyperbolic Equations ◽  
Differentiability Of Solutions

scholarly journals Cotton-Type and Joint Invariants for Linear Elliptic Systems

2013 ◽  
Vol 2013 ◽  
pp. 1-6 ◽  
Author(s):  
A. Aslam ◽  
F. M. Mahomed
Keyword(s):  
Elliptic Equations ◽  
Hyperbolic Equations ◽  
Elliptic Systems ◽  
General System ◽  
Linear Elliptic Equation ◽  
Linear Complex ◽  
Independent Variables ◽  
Joint Invariants ◽  
Laplace Type ◽  
Linear Hyperbolic Equations

Cotton-type invariants for a subclass of a system of two linear elliptic equations, obtainable from a complex base linear elliptic equation, are derived both by spliting of the corresponding complex Cotton invariants of the base complex equation and from the Laplace-type invariants of the system of linear hyperbolic equations equivalent to the system of linear elliptic equations via linear complex transformations of the independent variables. It is shown that Cotton-type invariants derived from these two approaches are identical. Furthermore, Cotton-type and joint invariants for a general system of two linear elliptic equations are also obtained from the Laplace-type and joint invariants for a system of two linear hyperbolic equations equivalent to the system of linear elliptic equations by complex changes of the independent variables. Examples are presented to illustrate the results.

scholarly journals Pointwise boundary differentiability of solutions of elliptic equations

2016 ◽  
Vol 151 (3-4) ◽  
pp. 469-476
Author(s):  
Yongpan Huang ◽  
Dongsheng Li ◽  
Kai Zhang
Keyword(s):  
Elliptic Equations ◽  
Differentiability Of Solutions

scholarly journals A regularity result for a class of elliptic equations with lower order terms

2020 ◽  
Vol 6 (2) ◽  
pp. 751-771 ◽  
Author(s):  
Claudia Capone ◽  
Teresa Radice
Keyword(s):  
Dirichlet Problem ◽  
Lower Order ◽  
Elliptic Equations ◽  
Order Term ◽  
Regularity Result ◽  
Lower Order Term ◽  
Bounded Solutions ◽  
Differentiability Of Solutions ◽  
Higher Differentiability ◽  
Lower Order Terms

Abstract In this paper we establish the higher differentiability of solutions to the Dirichlet problem $$\begin{aligned} {\left\{ \begin{array}{ll} \text {div} (A(x, Du)) + b(x)u(x)=f &{} \text {in}\, \Omega \\ u=0 &{} \text {on} \, \partial \Omega \end{array}\right. } \end{aligned}$$ div ( A ( x , D u ) ) + b ( x ) u ( x ) = f in Ω u = 0 on ∂ Ω under a Sobolev assumption on the partial map $$x \rightarrow A(x, \xi )$$ x → A ( x , ξ ) . The novelty here is that we take advantage from the regularizing effect of the lower order term to deal with bounded solutions.

scholarly journals Green's functions and Riemann's method

1965 ◽  
Vol 14 (4) ◽  
pp. 293-302 ◽  
Author(s):  
A. G. Mackie
Keyword(s):  
Fundamental Solution ◽  
Green’S Function ◽  
Boundary Value Problems ◽  
Green's Function ◽  
Elliptic Equations ◽  
Green's Functions ◽  
Green’S Functions ◽  
Hyperbolic Equations ◽  
Boundary Value ◽  
Auxiliary Function

Methods for solving boundary value problems in linear, second order, partial differential equations in two variables tend to be somewhat rigidly partitioned in some of the standard text-books. Problems for elliptic equations are sometimes solved by finding the fundamental solution which is defined as a solution with a given singularity at a certain point. Another approach is by way of Green's functions which are usually defined as solutions of the original homogeneous equations now made inhomogeneous by the introduction of adelta function on the right hand side. The Green's function coincides with the fundamental solution for elliptic equations but exhibits a totally different type of singularity for parabolic or hyperbolic equations. Boundary value problems for hyperbolic equations can often by solved by Riemann's method which depends on the existence of an auxiliary function called the Riemann or sometimes the Riemann-Green function. The main object of this paper is to show the close relationship between Riemann's method and the method of Green's functions. This not only serves to unify different methods of solution of boundary value problems but also provides an additional method of determining Riemann functions for given hyperbolic equations. Before establishing these relationships we shall survey the general approach to boundary value problems through the use of the Green's function.

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