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I am looking for the references or simple direct proofs of existence and uniqueness of viscosity solution for two problems:

$1.$ Let $\Omega$ be bounded domain, $u=0$ on $\delta\Omega$ and $$|Du|-f(x,u)=0$$ where $f\ge 0$ and $f(x,r)<f(x,s)$ for all $x\in \Omega$ and $r<s.$

$2.$ $-D\cdot a(Du)=f$ where $a$ is a smooth vector-valued function that satisfies

monotonicity condition $(a(Du)-a(Dv))\cdot (Du-Dv)\ge 0.$ We can assume whatever we need for the domain in $R^n,$ $u=0$ on the boundary.

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Welcome to Math.SE! Regarding 2, you can't get uniqueness without strict monotonicity. Indeed, if $a(p_1,p_2)=(-p_2,p_1)$ in two dimensions, then $D\cdot a(Du)=0$ holds for any smooth function at all. – user53153 Jan 14 '13 at 6:10
You are right, thanks. For my purposes, we can assume pretty much everything. So let it be strictly monotone. – Alvin Jan 14 '13 at 10:08
Cross-posted to MathOverflow – user53153 Jan 14 '13 at 16:23
up vote 1 down vote accepted

Edit: the first part works only if $u_1$ and $u_2$ are differentiable.

For the first question, what you really need to prove is comparison result, namely if $u_1$ and $u_2$ are corresponding viscosity sub and super solutions that agree on the boundary, then $u_1\le u_2.$ For this purpose, fix $x_0\in\Omega$ and take $\phi\in C^2(\Omega),$ such that $u_2-\phi$ attains local minimum at $x_0.$ Since, $u_2$ is a super solution, it follows that $$|D\phi(x_0)|\le f(x,u_2(x_0)).$$

By interpolation theorem, for each value of $p$ there exists $C^2$ function $\phi_1,$ such that $u_1-\phi$ attains local minimum at $x_0$ and $|D\phi_(x_0)|=p=|D\phi(x_0)|.$ For such defined $\phi$ and $\phi_1,$ recalling that $u_1$ is a sub solution, we get $$f(x,u_1(x_0))\le |D\phi_1(x_0)|=|D\phi(x_0)|\le f(x,u_2(x_0)),$$ and the result follows from monotonicity condition.

For Problem 2, see:

Ishii, H.; Lions, P.-L. Viscosity solutions of fully nonlinear second-order elliptic partial differential equations. J. Differential Equations 83 (1990), no. 1, 26–78. (Reviewer: Philippe Delanoë) 35Bxx (35J60)

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