Complex Analysis : Exercises With Solutions

This text contains the solutions to all of the practice problems in the 10th chapter of the lecture notes “An Introduction to Complex Analysis”.OstravaE-vyuk

Resonance problems for p-Laplacian

We study the existence of the weak solution of the nonlinear boundary value problem [GRAPHICS] where p and lambda are real numbers, p > 1, h is an element of L-P'(0, pi)(p' = p/(p - 1)) and the nonlinearity g : R --> R is a continuous function of the Landesman-Lazer type. Our results generalize previously published results about the solvability of our problem

Strong resonance problems for the one-dimensional p-Laplacian

We study the existence of the weak solution of the nonlinear boundary-value problem -(vertical bar u'vertical bar(p-2)u')' = lambda vertical bar u vertical bar(p-2)u + g(u) - h(x) in (0, pi), u(0) = u(pi) = 0, where p and lambda are real numbers, p > 1, h is an element of L-p' (0, pi) (p' = p/p-1) and the nonlinearity g : R -> R is a continuous function of the Landesman-Lazer type. Our sufficiency conditions generalize the results published previously about the solvability of this problem.Web of Scienceart. no. 0

Věty o minimaxu a jejich použití

Import 20/04/2006Prezenční výpůjčkaVŠB - Technická univerzita Ostrava. Fakulta elektrotechniky a informatiky

A new approach to solving a quasilinear boundary value problem with pp-Laplacian using optimization

summary:We present a novel approach to solving a specific type of quasilinear boundary value problem with $p$-Laplacian that can be considered an alternative to the classic approach based on the mountain pass theorem. We introduce a new way of proving the existence of nontrivial weak solutions. We show that the nontrivial solutions of the problem are related to critical points of a certain functional different from the energy functional, and some solutions correspond to its minimum. This idea is new even for $p=2$. We present an algorithm based on the introduced theory and apply it to the given problem. The algorithm is illustrated by numerical experiments and compared with the classic approach

Shape optimization and subdivision surface based approach to solving 3D Bernoulli problems

In the paper we consider a treatment of Bernoulli type shape optimization problems in three dimensions by the combination of the boundary element method and the hierarchical algorithm based on the subdivision surfaces. After proving the existence of the solution on the continuous level we discretize the free part of the surface by a hierarchy of control meshes allowing to separate the mesh necessary for the numerical analysis and the choice of design parameters. During the optimization procedure the mesh is updated starting from its coarse representation and refined by adding design variables on finer levels. This approach serves as a globalization strategy and prevents geometry oscillations without any need for remeshing. We present numerical experiments demonstrating the capabilities of the proposed algorithm.Web of Science7892932291