Abstract
We consider parabolic operators of the form $$\partial_t+\mathcal{L}, \mathcal{L}:=-\mbox{div}\, A(X,t)\nabla,$$ in$\mathbb R_+^{n+2}:=\{(X,t)=(x,x_{n+1},t)\in \mathbb R^{n}\times \mathbb R\times ...
Highlights
Introduction and statement of main resultsIn this paper we establish certain estimates related to the solvability of the Dirichlet, Neumann and Regularity problems with data in L2, in the following these problems are referred to as (D2),(N2) and (R2), by way of layer potentials and for second order parabolic equations of the form (1.1)Hu := (∂t + L)u = 0, where n+1L := −div A(X, t)∇ = − ∂xi(Ai, j(X, t)∂xj) i, j=1 is defined in Rn+2 = {(X, t) = (x1, .., xn+1, t) ∈ Rn+1 × R}, n ≥ 1
In this paper we establish certain estimates related to the solvability of the Dirichlet, Neumann and Regularity problems with data in L2, in the following these problems are referred to as (D2), (N2) and (R2), by way of layer potentials and for second order parabolic equations of the form
In this paper we first prove, in the case of equations of the form (1.1), satisfying (1.2)-(1.3) and the De Giorgi-Moser-Nash estimates stated in (2.6)-(2.7) below, that a set of key boundedness estimates for associated single layer potentials can be reduced to two crucial estimates (Theorem 1.5), one being a square function estimate involving the single layer potential
Summary
In this paper we establish certain estimates related to the solvability of the Dirichlet, Neumann and Regularity problems with data in L2, in the following these problems are referred to as (D2),. One major contribution of these papers, see [HL], [H] and [HL1] in particular, is the proof of Theorem 1.5 in the context of the heat operator in time-dependent Lipschitz type domains Beyond these results the literature only contains modest contributions to the study of parabolic layer potentials associated to second order parabolic operators (in divergence form) with variable, bounded, measurable, uniformly elliptic (and complex) coefficients. Based on this we believe that our results will pave the way for important developments in the area of parabolic PDEs. While Theorem 1.5 and Theorem 1.8 coincide, in the stationary case, with the set up and the corresponding results established in [AAAHK] for elliptic equations, we claim that our results, Theorem 1.5 in particular, are not, for at least two reasons, straightforward generalizations of the corresponding results in [AAAHK]. We believe that our proof of Theorem 1.9 adds to the clarity of the corresponding argument in [FS]
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