Questions tagged [viscosity-solutions]

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2 votes
1 answer
186 views

Strategy of the proof of the "minimal entropy condition" for scalar conservation laws

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 ...
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8 votes
1 answer
184 views

Viscous approximation of Eikonal equation

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

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 \...
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0 votes
0 answers
70 views

Is a $C^{1,1}$ function a viscosity solution?

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 $C^{1,1}$...
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2 votes
1 answer
121 views

Ramp and Cliff Solutions to the Viscous Burgers Equation: Explicit Formula?

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

Concepts of Solutions to Partial Differential Equations

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 ...
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3 votes
1 answer
227 views

Reference request on Pucci extremal operators

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

Changing a little assumptions in famous paper Vanishing viscosity solutions of nonlinear hyperbolic systems?

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 ...
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0 votes
1 answer
171 views

Two types of limits of viscosity solutions

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

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) & \...
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4 votes
1 answer
609 views

Evans-Krylov theorem

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

Comparison principle for viscosity solution

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 ...
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3 votes
1 answer
107 views

Meaningful generalization of viscosity solutions to higher order equations

Is there a meaningful generalization of the notion of viscosity solutions to third and fourth order equations?
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3 votes
2 answers
569 views

Uniqueness of viscosity solutions of Hamilton-Jacobi equation

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 ...
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3 votes
0 answers
127 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]} \{...
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4 votes
2 answers
677 views

If a PDE has a unique classical solution, must it have a unique viscosity solution?

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 ...
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6 votes
1 answer
761 views

A question about the $C^{2,\alpha}$ regularity of concave fully nonlinear uniformly elliptic equation

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 ...
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4 votes
1 answer
405 views

regularity for viscosity solutions of second order parabolic equations

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

Why are viscosity solutions useful solutions?

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. ...
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