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Questions tagged [viscosity-solutions]

For questions on the definition, properties and applications of viscosity solutions.

The viscosity solution is a generalization of the classical concept of what is meant by a 'solution' to a partial differential equation . The viscosity solution is the natural solution concept to use in many applications of PDE's, including first order equations arising in optimal control (the Hamilton–Jacobi equation), differential games (the Isaacs equation) or front evolution problems.

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User's guide to viscosity solutions basic question

Page 9 of the user's guide has a claim that if $u$ is twice differentiable at $\hat{x}$ and $$ u\left(x\right)\leq u\left(\hat{x}\right)+\left\langle p,x-\hat{x}\right\rangle +\frac{1}{2}\left\langle X\left(x-\hat{x}\right),x-\hat{x}\right\rangle…
user194699
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Different conditions in the definition of viscosity solutions

Let $u:\Omega\to\mathbb{R}$ be continuous and $\Omega\subset\mathbb{R}^n$. I've lately seen a definition of viscosity solutions where the two conditions for the test functions in a point $x_0\in\Omega$ was: $\phi\in C^2(\mathbb{R}^n)$ such that…
HelloEveryone
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The little push in the graph "preserves" infimum

i have a limited domain $X \subset \mathbb{R}^{n}$, and two functions, $u$ continuous in $X$ and $ f \in C^{2}(X)$, such that $u-f$ has local minimum in a given $x_{0} \in X$, ie for some $p> 0$, $x_{0}$ minimizes $u-f$ in ball with center…
Cézar Bezerra
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Understanding the definition of viscosity solutions

I am thinking about the definition of viscosity solutions. One is interested in pde's of the form $F(x,u(x),Du(x),D^2u(x))=0$ in $\Omega$ for some $\Omega\subset\mathbb{R}^n$. Later in most books there arises at some point the discussion of…
HelloEveryone
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