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The \(L^ p\)-integrability of Green’s functions and fundamental solutions for elliptic and parabolic equations. (English) Zbl 0567.35003

Let be \(L_ au=\sum^{d}_{i,j=1}a^{ij}(x)\partial^ 2u/\partial x_ i\partial x_ j\) where \(d\geq 1\) and \(a\equiv (a^{ij}(x))\) is a smooth, symmetric, \(d\times d\)-matrix-valued function on \({\mathbb{R}}^ d\) satisfying, for given \(\lambda\in (0,1)\), \(\lambda I\leq a(x)\leq \lambda^{-1}I\) (in the sense of nonnegative definiteness).
The present paper mainly considers three topics. First, for the equation \(L^*_ av=0\) where \(L^*_ a\) denotes the formal adjoint of \(L_ a\), in a bounded domain \(\Omega \subset {\mathbb{R}}^ d\) the interior behavior of nonnegative solutions v is investigated. In detail, for these solutions a backward or reversed Hölder inequality is proved for suitable balls in \(\Omega\). This result is applied to the Green’s function corresponding to \(L_ a\) and \(\Omega\).
Secondly, a direct proof of Harnack’s inequality for nonnegative solutions u of \(L_ au=0\) in \(\Omega\) is given. Here, essentially ideas by M. V. Safonov [Zap. Nauchn. Semin. Leningr. Otd. Mat. Inst. Steklova 96, 272-287 (1980; Zbl 0458.35028)] and N. S. Trudinger [Invent. Math. 61, 67-79 (1980; Zbl 0453.35028)] are utilized.
Finally, for parabolic equations of the form \(\partial u/\partial t=L_ au\) \((L_ a\) as above) in \((0,\infty)\times {\mathbb{R}}^ d\), the \(L^ p\)- integrability of fundamental solutions is investigated in the case of the Cauchy problem with data given on \(\{0\}\times {\mathbb{R}}^ d\).
Reviewer: M.Kracht

MSC:

35A08 Fundamental solutions to PDEs
35J25 Boundary value problems for second-order elliptic equations
35K15 Initial value problems for second-order parabolic equations
35B05 Oscillation, zeros of solutions, mean value theorems, etc. in context of PDEs
35C15 Integral representations of solutions to PDEs
35B35 Stability in context of PDEs
35B45 A priori estimates in context of PDEs
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