Questions tagged [viscosity-solutions]

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

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About viscosity solutions of elliptic equations

i am recently starting my studies on solutions of elliptic pde's of second order in the sense of viscosity. From this study some doubts result, they are: What is the motivation to introduce a new ...
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Unique Uniformily Viscosity Solution

Say we have the following pocess: $$dX^{\epsilon} = b(X^{\epsilon}(s))ds + \sqrt\epsilon \sigma(X^{\epsilon}(s))dW(s)$$ for $s \in [0, T]$ and $X^{\epsilon}(0) = x_0$ We want to estimate a quantity ...
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Introduction to viscosity solutions theory

Can you recommend an introduction to viscosity solutions theory? More specifically, I'm looking for a modern treatment similar to Chapter 10 of Evans's Partial Differential Equations, but somewhat ...
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In the User's Guide to Viscosity Solutions, there is an example (Section 5.D; page 30) using the Crandall-Ishii lemma on unbounded domains. Equation (5.17) claims $$-3\alpha\left(\begin{array}{cc} I\... 0answers 51 views Viscosity solution in for a non smooth value function I am trying to find a solution for a free boundary problem, where the value function won't be smooth as it is in a multi period framework with fixed costs at each period. Resulting jumps in the ... 1answer 877 views Viscous Burgers' equation using Lax-Wendroff scheme I was assigned to find a solution for the Burgers' equation$$u_t+uu_x=\upsilon u_{xx}$$with initial condition u(x,0) = \sin(x) and with boundary conditions u(0,t)=u(\epsilon,t) with \epsilon &... 0answers 132 views To show a given function is not the viscosity solution. For the equation  F(x,u,u',u'') = -au''-1 =0 for  x\in (0,2) with  u(0) = 0 = u(2)  and a(x) is 1 for x\in (0,1) and 2 for x\in [1,2). Need to show that the function$$ u(x) = \begin{...
Let $u^i$, $i=1,2$, be viscosity solutions of \begin{align*} u_t^i + H(Du^i,x) & = 0\quad\mathrm{in}\ \mathbb R^n\times (0,\infty)\\ u^i & = g^i\quad\mathrm{on}\ \mathbb R^n\times \{t=0\} \...