scholarly journals On Strongly Nonlinear Poincaré Boundary Value Problems for Harmonic Functions

1984 ◽  
Vol 3 (5) ◽  
pp. 385-399 ◽  
Author(s):  
Lothar von Wolfersdorf
Keyword(s):  
Boundary Value Problems ◽  
Harmonic Functions ◽  
Boundary Value ◽  
Strongly Nonlinear

Logarithmic potential and generalized analytic functions

2021 ◽  
Vol 18 (1) ◽  
pp. 12-36
Author(s):  
Vladimir Gutlyanskii ◽  
Olga Nesmelova ◽  
Vladimir Ryazanov ◽  
Artyem Yefimushkin
Keyword(s):  
Boundary Value Problems ◽  
Analytic Functions ◽  
Harmonic Functions ◽  
Boundary Data ◽  
Boundary Value ◽  
Logarithmic Capacity ◽  
Functional Interpretation ◽  
Arbitrary Boundary ◽  
Generalized Analytic Functions ◽  
Complex Valued

The study of the Dirichlet problem in the unit disk $\mathbb D$ with arbitrary measurable data for harmonic functions is due to the famous dissertation of Luzin [31]. Later on, the known monograph of Vekua \cite{Ve} has been devoted to boundary-value problems (only with H\"older continuous data) for the generalized analytic functions, i.e., continuous complex valued functions $h(z)$ of the complex variable $z=x+iy$ with generalized first partial derivatives by Sobolev satisfying equations of the form $\partial_{\bar z}h\, +\, ah\, +\ b{\overline h}\, =\, c\, ,$ where it was assumed that the complex valued functions $a,b$ and $c$ belong to the class $L^{p}$ with some $p>2$ in smooth enough domains $D$ in $\mathbb C$. The present paper is a natural continuation of our previous articles on the Riemann, Hilbert, Dirichlet, Poincar\'{e} and, in particular, Neumann boundary-value problems for quasiconformal, analytic, harmonic, and the so-called $A-$harmonic functions with boundary data that are measurable with respect to logarithmic capacity. Here, we extend the corresponding results to the generalized analytic functions $h:D\to\mathbb C$ with the sources $g$ : $\partial_{\bar z}h\ =\ g\in L^p$, $p>2\,$, and to generalized harmonic functions $U$ with sources $G$ : $\triangle\, U=G\in L^p$, $p>2\,$. This paper contains various theorems on the existence of nonclassical solutions of the Riemann and Hilbert boundary-value problems with arbitrary measurable (with respect to logarithmic capacity) data for generalized analytic functions with sources. Our approach is based on the geometric (theoretic-functional) interpretation of boundary-values in comparison with the classical operator approach in PDE. On this basis, it is established the corresponding existence theorems for the Poincar\'{e} problem on directional derivatives and, in particular, for the Neumann problem to the Poisson equations $\triangle\, U=G$ with arbitrary boundary data that are measurable with respect to logarithmic capacity. These results can be also applied to semi-linear equations of mathematical physics in anisotropic and inhomogeneous media.

scholarly journals Boundary value problems associated with singular strongly nonlinear equations with functional terms

2020 ◽  
Vol 10 (1) ◽  
pp. 684-706
Author(s):  
Stefano Biagi ◽  
Alessandro Calamai ◽  
Cristina Marcelli ◽  
Francesca Papalini
Keyword(s):  
Boundary Value Problems ◽  
Existence Result ◽  
Nonlinear Differential Equations ◽  
Boundary Value ◽  
Upper And Lower Solutions ◽  
Weak Sense ◽  
Functional Term ◽  
Strongly Nonlinear ◽  
Negative Function ◽  
Null Measure

Abstract We study boundary value problems associated with singular, strongly nonlinear differential equations with functional terms of type $$\big({\it \Phi}(k(t)\,x'(t))\big)' + f(t,{{\mathcal{G}}}_x(t))\,\rho(t, x'(t)) = 0,$$ on a compact interval [a, b]. These equations are quite general due to the presence of a strictly increasing homeomorphism Φ, the so-called Φ-Laplace operator, of a non-negative function k, which may vanish on a set of null measure, and moreover of a functional term Gx. We look for solutions, in a suitable weak sense, which belong to the Sobolev space W1,1([a, b]). Under the assumptions of the existence of a well-ordered pair of upper and lower solutions and of a suitable Nagumo-type growth condition, we prove an existence result by means of fixed point arguments.

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