Questions tagged [viscosity-solutions]

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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 ...
5
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1answer
170 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
390 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
128 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
99 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?
2
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2answers
446 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 ...
3
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0answers
113 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
551 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
636 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
377 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 ...
20
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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. ...