Questions tagged [viscosity-solutions]

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Non-linearity of viscosity solutions

I am interested in the following problem. Let consider the solution of the non-linear PDE on $[0,T]\times\mathbb{R}$ satifying the following Cauchy problem: $$ \begin{cases} u_t = F(u_{xx}),\\ u(0,x) =...
NancyBoy's user avatar
  • 175
2 votes
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Smooth dependence of parameter of PDE - viscosity solutions

There is a prevalent method called the "Nonlinear adjoint method" in the study of viscosity solution and Hamilton--Jacobi equation, especially equations of the form $$ u^\varepsilon + H(x,Du^...
Sean's user avatar
  • 313
1 vote
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Viscosity characterization of convex functions

Let $\Omega\subseteq\mathbb{R}^n$ open and convex. It is elementary that if $u\in C^2(\Omega)$ then $$u \text{ is convex}\iff D^2u\geq0 \ \text{ in } \Omega$$ I was looking for a similar ...
Luca.b's user avatar
  • 113
3 votes
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What notion of weak solution is suitable for systems of $\infty$-elliptic PDE?

Let $Pu = f$ be an elliptic PDE in divergence form. Then $P$ is viewed as a generalization of the Laplacian, and we can define its weak solutions analogously to how we define a weakly harmonic ...
Aidan Backus's user avatar
3 votes
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Approximation of viscosity subsolution

Let $u: \Omega \to \mathbf{R}$ be a $C^{0,\alpha}$ function, with $\alpha \in (0,1]$, defined on a bounded, open domain $\Omega$. Suppose that $u$ is a viscosity subsolution of the equation $\Delta U =...
Leo Moos's user avatar
  • 4,912
2 votes
1 answer

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 ...
user avatar
8 votes
1 answer

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 ...
user483557's user avatar
2 votes
1 answer

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 \...
Zac's user avatar
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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 $C^{1,1}$...
ymh's user avatar
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2 votes
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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 ...
Gateau au fromage's user avatar
5 votes
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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 ...
Peter Wacken's user avatar
3 votes
1 answer

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{...
Jingeon An-Lacroix's user avatar
2 votes
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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 ...
Mark's user avatar
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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 ...
Jan Lynn's user avatar
  • 101
4 votes
1 answer

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) & \...
user avatar
4 votes
1 answer

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 ...
user128943's user avatar
3 votes
1 answer

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. I ...
mnmn1993's user avatar
3 votes
1 answer

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?
user avatar
3 votes
2 answers

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 ...
user avatar
3 votes
0 answers

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]} \{...
kenneth's user avatar
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4 votes
2 answers

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 ...
lost1's user avatar
  • 373
6 votes
1 answer

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 ...
Thomas's user avatar
  • 71
4 votes
1 answer

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 ...
saurabh trivedi's user avatar
26 votes
1 answer

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. ...
shuhalo's user avatar
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