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General existence theorems for Hamilton-Jacobi equations in the scalar and vectorial cases. (English) Zbl 0901.49027
Consider the following Dirichlet problem for the Hamilton-Jacobi equation: $F(Du(x))= 0,\qquad\text{a.e. }x\in\Omega,$ $u(x)= \varphi(x),\qquad x\in\partial\Omega,$ where $$\Omega$$ is an open bounded set of $$\mathbb{R}^n$$, $$F: \mathbb{R}^{n\times N}\to \mathbb{R}$$ and $$\varphi\in W^{1,\infty}(\Omega,\mathbb{R}^N)$$, with $$n,N\geq 1$$.
In this paper, the authors introduce some new and interesting conditions on $$F$$ and $$\varphi$$ and a new method to prove existence of solutions.
In the scalar case, $$N= 1$$, they prove that if $$E= \{z\in\mathbb{R}^n: F(z)= 0\}$$ is closed and $D\varphi\in E\cup\text{int co }E,$ where $$\text{int co }E$$ denotes the interior of convex hull of $$E$$, the problem has solutions in $$W^{1,\infty}$$.
In the vectorial case $$N>1$$, the existence proof is achieved provided that $$F$$ is quasiconvex and satisfies some additional assumptions.
Finally, the prescribed singular values case is also studied. The proofs use the Baire category methods and weak lower semicontinuity arguments.

##### MSC:
 49L25 Viscosity solutions to Hamilton-Jacobi equations in optimal control and differential games
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##### References:
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