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\(C^{1,\beta}\) regularity of viscosity solutions via a continuous-dependence result. (English) Zbl 1107.35032

The author is interested in the regularity of viscosity solutions of fully nonlinear parabolic equations, namely
\[ u_t-F(D^2u,Du,u,x,t)= f(x, t)\quad\text{in }\mathbb{R}^N\times (0,t),\tag{1} \]
\[ u(x,0)=u_0(x)\quad \text{in }\mathbb{R}^N,\tag{2} \]
where \(F\) is a continuous, real-valued function defined on \(S^n\times\mathbb{R}^N \times\mathbb{R}\times\mathbb{R}^N \times[0,T]\), \(S^N\) being the space of \(N\times N\) symmetric matrices equipped with usual ordering, and \(f\) is a continuous, real-valued function defined on \(\mathbb{R}^N\times[0,T]\). The contribution of this paper is to show the existence of viscosity solutions of (1)–(2), namely, solutions which are locally \(C^{1,\beta}\) or \(C^{2, \beta}\) in \(x\), the regularity properties in \(t\) being then either just a consequence of the regularity properties in \(x\) or proved simultaneously. For to prove the existence and uniqueness of continuous solutions of (1)–(2), together with their local Lipschitz continuity in \(x\) and local Hölder continuity in \(t\), the key steps are on one hand a comparison result for viscosity solutions and on the other hand a continuous-dependence result, which are both valid for strictly subquadratic viscosity solutions of (1)–(2).

MSC:

35B65 Smoothness and regularity of solutions to PDEs
35B50 Maximum principles in context of PDEs
35K55 Nonlinear parabolic equations
35K15 Initial value problems for second-order parabolic equations
35A05 General existence and uniqueness theorems (PDE) (MSC2000)
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