Questions tagged [viscosity-solutions]

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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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1answer
80 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 ...
3
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75 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 ...
2
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1answer
107 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
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0answers
67 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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1answer
148 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
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1answer
190 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) & \...
3
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1answer
480 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
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1answer
146 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 ...
3
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1answer
102 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?
3
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2answers
511 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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123 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]} \{...
4
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2answers
610 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 ...
6
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1answer
711 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 ...
4
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1answer
389 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 ...
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1answer
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. ...