# Questions tagged [viscosity-solutions]

The viscosity-solutions tag has no usage guidance.

The viscosity-solutions tag has no usage guidance.

19
questions

2
votes

1
answer

186
views

Combining Theorem 2.3 and Corollary 2.5 of this paper gives that, for a strictly convex conservation law
$$u_t + f(u)_x = 0,$$ satisfying the entropy condition
$$\eta(u)_t + q(u)_x \le 0$$ in the ...

8
votes

1
answer

184
views

Consider the Eikonal equation
\begin{align*}
\begin{cases}\left|D u\right|^{2}=1 & \text { on } \Omega \\ u \equiv 0 & \text { on } \partial \Omega\end{cases}
\end{align*}
and the viscous ...

2
votes

1
answer

178
views

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 \...

0
votes

0
answers

70
views

Assume $F(A)$ is a degenerate elliptic operator. Let $u$ be the limit of a smooth sequence $({u_n})$ and $F(D^2 u_n)=f_n\ge 0$ with $(f_n)$ converging uniformly to $0$. If we know that $u$ is $C^{1,1}$...

2
votes

1
answer

121
views

I read an article in which the authors describe an observed phenomenon as being related to the "classical ramp and cliff Burgers solutions''. Those are described as Burgers solutions that behave ...

4
votes

0
answers

94
views

I already asked this question on math stackexchange (see here), but since I didn't get an answer there, I was wondering if I would be more lucky here.
I was wondering what the most used notions for ...

3
votes

1
answer

227
views

While reading [1], I encountered with the concept "Pucci extremal operator" which is defined by:
$$M_\Lambda^-(N):=\left(\sum\text{positive eigenvalues of }N\right)+\Lambda\left(\sum\text{...

2
votes

0
answers

74
views

The question that I hope to find some answer here is: do the results from
Bianchini, Bressan, Vanishing viscosity solutions of nonlinear hyperbolic systems, 2005
paper still apply if we change a ...

0
votes

1
answer

171
views

I actually posted this on math.stackexchange but it wasn't getting responses even after a bounty. I thought maybe it is too specialized so I'll post it here. I'm currently reading the user's guide to ...

4
votes

1
answer

232
views

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) & \...

4
votes

1
answer

609
views

Do there exist estimates for nonconcave functionals similar to Evans-Krylov theorem in chapter 6 of Fully nonlinear elliptic equations by Luis A.C affarelli and Cabre? Perhaps there is a ...

2
votes

1
answer

190
views

I am currently reading the paper "The Inverse Mean Curvature Flow and the Riemannian Penrose Inequality" written by Gerhard Huisken and Tom Ilmanen.
https://projecteuclid.org/euclid.jdg/1090349447
I ...

3
votes

1
answer

107
views

Is there a meaningful generalization of the notion of viscosity solutions to third and fourth order equations?

3
votes

2
answers

569
views

Consider the following Hamilton-Jacobi (HJ) equation:
$$u_t + H(\nabla u,x) = 0 \quad \text{ in } \mathbb{R}^n \times (0, T], $$ where $u:\mathbb{R}^n \times (0,T] \to \mathbb{R}$, and $H:\mathbb{R}^n ...

3
votes

0
answers

127
views

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]} \{...

4
votes

2
answers

677
views

If a PDE has a unique classical solution, must it have a unique viscosity solution?
The particular problem I am interested in is parabolic, but I would be interested in the general case.
A short ...

6
votes

1
answer

761
views

While reading Theorem 6.6 of Chapter Six of "Fully nonlinear elliptic equation" by Luis A. Caffarelli and Xavier Cabre in the American mathematical society colloquium publications vol. 43, I get two ...

4
votes

1
answer

405
views

I would like to know whether viscosity solutions to $u_{t} - F( D^{2} (u) ) = 0$ are $C^{1, \alpha}$ analogous to the elliptic case as in the book by Caffarelli and Cabre .
Here F is ...

23
votes

1
answer

3k
views

I refer to definition of viscosity solution in user's guide to viscosity solutions of second order partial differential equations by Michael G. Crandall, Hitoshi Ishii and Pierre-Louis Lions.
...