We discuss the existence of weak solutions to one class of Neumann boundary value problems (BVP) for non-linear elliptic equation. Because of the specific of nonlinearity, we cannot a priori expect to have a solution in the standard functional space. Instead of this we show that the original BVP admits the so-called approximate weak solution. To do so, we introduce a special family of perturbed optimal control problems (OCPs) where the class of fictitious controls are closely related with the properties of distribution in right-hand side of the elliptic equation, and we show that optimal solutions of such problems allow to attain (in the limit) some approximate solutions as the parameter of perturbation $\e>0$ tends to zero. The main questions we discuss in this paper touch on solvability of perturbed OCPs, uniqueness of their solutions, asymptotic properties of optimal pairs as the perturbation parameter $\e>0$ tends to zero, and deriving of optimality conditions for the perturbed OCPs. As a consequence, we obtain the sufficient conditions of the existence of weak solutions to the given class of nonlinear Neumann BVP and propose the way for their approximation.
On Approximate Solutions to the Neumann Elliptic Boundary Value Problem with non Linearity of Exponential Type
MANZO, Rosanna;
2016-01-01
Abstract
We discuss the existence of weak solutions to one class of Neumann boundary value problems (BVP) for non-linear elliptic equation. Because of the specific of nonlinearity, we cannot a priori expect to have a solution in the standard functional space. Instead of this we show that the original BVP admits the so-called approximate weak solution. To do so, we introduce a special family of perturbed optimal control problems (OCPs) where the class of fictitious controls are closely related with the properties of distribution in right-hand side of the elliptic equation, and we show that optimal solutions of such problems allow to attain (in the limit) some approximate solutions as the parameter of perturbation $\e>0$ tends to zero. The main questions we discuss in this paper touch on solvability of perturbed OCPs, uniqueness of their solutions, asymptotic properties of optimal pairs as the perturbation parameter $\e>0$ tends to zero, and deriving of optimality conditions for the perturbed OCPs. As a consequence, we obtain the sufficient conditions of the existence of weak solutions to the given class of nonlinear Neumann BVP and propose the way for their approximation.I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.
