Search results for " Dirichlet problem"
showing 3 items of 13 documents
Rotationally symmetric 1-harmonic flows from D2 TO S 2: Local well-posedness and finite time blowup
2010
The 1-harmonic flow from the disk to the sphere with constant Dirichlet boundary conditions is analyzed in the case of rotational symmetry. Sufficient conditions on the initial datum are given, such that a unique classical solution exists for short times. Also, a sharp criterion on the boundary condition is identified, such that any classical solution will blow up in finite time. Finally, nongeneric examples of finite time blowup are exhibited for any boundary condition.
Multi-parameter analysis of the obstacle scattering problem
2022
Abstract We consider the acoustic field scattered by a bounded impenetrable obstacle and we study its dependence upon a certain set of parameters. As usual, the problem is modeled by an exterior Dirichlet problem for the Helmholtz equation Δu + k 2 u = 0. We show that the solution u and its far field pattern u ∞ depend real analytically on the shape of the obstacle, the wave number k, and the Dirichlet datum. We also prove a similar result for the corresponding Dirichlet-to-Neumann map.
Variational versus pseudomonotone operator approach in parameter-dependent nonlinear elliptic problems
2013
We study the existence of nontrivial solutions of parameter-dependent quasilinear elliptic Dirichlet problems of the form $-\Delta u = \lambda f(u)$ in $\Omega$, $u = 0$ on $\partial\Omega$, in a bounded domain $\Omega$ with sufficiently smooth boundary, where $\lambda$ is a real parameter and $\Delta_p$ denotes the p-Laplacian. Recently the authors obtained multiplicity results by employing an abstract localization principle of critical points of functional of the form $\Phi-\lambda\Psi$ on open subleveis of $\Phi$, i.e., of sets of the form $\Phi^{-1}(-\infty,r)$, combined with differential inequality techniques and topological arguments. Unlike in those recent papers by the authors, the …