All Questions
3 questions
4
votes
1
answer
370
views
Equivalence of viscosity and weak solutions for the Poisson equation
Suppose $\Omega$ is a bounded smooth domain in $\mathbb{R}^d$.
How does one prove that weak solutions are viscosity solutions and vice versa for the problem
$$
\begin{cases}
-\Delta u = f(x) & \...
user123456
3
votes
0
answers
140
views
Sufficient condition for the unique solvability of Dirichlet problem of Hamilton-Jacobi equation
It shall be an old story in PDE.
I am looking for a sufficient condition of Dirichlet problem for the existence of the unique viscosity solution of the equation in the form of
$$\inf_{a \in [-1,1]} \{...
- 1,399
2
votes
1
answer
196
views
Viscosity solutions of $(-\Delta)^s u = 0$ in $\Omega $ with non-homogeneous data $u = 1$ in $\mathbb R^n \setminus \Omega$
Let us consider a smooth bounded domain $\Omega \subset \mathbb R^n$ and the problem
$$
(1) \quad \begin{cases}
(-\Delta)^s u +\lambda u= 0 & x \in \Omega \\
u = 1 & x \in \mathbb R^n \...
- 161