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Hölder estimates of solutions of singular parabolic equations with measurable coefficients. (English) Zbl 0836.35029
Let \(\Omega\) be a domain in \(\mathbb{R}^N\), and for \(0 < T < \infty\) set \(\Omega_T \equiv \Omega \times (0,T)\). Consider singular parabolic equations of the type \[ u_t - \bigl( a_{ij} (x,t) |Du |^{p - 2} u_{x_i} \bigr)_{x_j} = 0 \quad \text{in } \Omega_T,\;1 < p < 2,\tag{1} \] where the entries of the matrix \((a_{ij})\) are only measurable and satisfy the ellipticity condition \[ \Lambda^{-1} |\xi |^2 \leq a_{ij} (x,t) \xi_i \xi_j \leq \Lambda |\xi |^2,\;\forall \xi \in \mathbb{R}^N, \quad \text{a.e. } (x,t) \in \Omega_T, \] for some \(\Lambda > 1\). The equation is singular since the modulus of ellipticity blows up at points where \(|Du |= 0\). One result of this paper is that weak solutions \[ u \in C_{\text{loc}} \bigl( 0,T; L^2_{\text{loc}} (\Omega) \bigr) \cap L^p_{\text{loc}} \bigl( 0,T; W^{1,p}_{\text{loc}} (\Omega) \bigr) \cap L^\infty_{\text{loc}} (\Omega_T) \] are locally Hölder continuous in \(\Omega_T\). We introduce a novel iteration technique which we feel is of independent interest.

35D10 Regularity of generalized solutions of PDE (MSC2000)
35K65 Degenerate parabolic equations
Full Text: DOI
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