The method of layer potentials for the heat equation in time-varying domains. (English) Zbl 0826.35041
Mem. Am. Math. Soc. 545, 157 p. (1995).
The authors want to study parabolic boundary value problems by the classical method of layer potentials. Generalizing earlier works of R. M. Brown [Am. J. Math. 111, No. 2, 339-379 (1989; Zbl 0696.35065)] concerning the heat equation in Lipschitz cylinders, and R. Kaufman and J.-M. Wu [Compos. Math. 65, No. 2, 201-207 (1988; Zbl 0651.31002)] concerning domains with time dependent boundaries in the case of a single space dimension, the authors consider in this memoir the noncylindrical higher dimensional case.
With respect to other results of J. T. Kemper [Trans. Am. Math. Soc. 167, 243-262 (1972; Zbl 0238.35039)], valid for more general parabolic equations and systems, the use of the layer potentials allows here \(L^p\)-theory under weak hypotheses on the boundary of the domain. Precisely, the authors assume the boundary is given by the graph of a function which is \(\text{Lip}_1\) in the space variables and belongs to a half order BMO Sobolev space relative to the time variable.
The memoir consists of three chapters. Chapter 1 is devoted to prove the \(L^p\)-boundedness of the singular integral operators related to double layer heat potentials, by using methods of Coifman, Meyer and Lemaire. Let us observe that the generality of the assumptions on the boundary introduces a particular complexity into the problem. Then, in Chapter 2, the authors develop a modification of the scheme of G. David [Ann. Sci. Éc. Norm. Supér, IV. Sér. 21, No. 2, 225-258 (1988; Zbl 0655.42013)] to deduce \(L^p\)-boundedness of the double layer potential. In Chapter 3 applications are given to the solutions of the boundary value problem.
With respect to other results of J. T. Kemper [Trans. Am. Math. Soc. 167, 243-262 (1972; Zbl 0238.35039)], valid for more general parabolic equations and systems, the use of the layer potentials allows here \(L^p\)-theory under weak hypotheses on the boundary of the domain. Precisely, the authors assume the boundary is given by the graph of a function which is \(\text{Lip}_1\) in the space variables and belongs to a half order BMO Sobolev space relative to the time variable.
The memoir consists of three chapters. Chapter 1 is devoted to prove the \(L^p\)-boundedness of the singular integral operators related to double layer heat potentials, by using methods of Coifman, Meyer and Lemaire. Let us observe that the generality of the assumptions on the boundary introduces a particular complexity into the problem. Then, in Chapter 2, the authors develop a modification of the scheme of G. David [Ann. Sci. Éc. Norm. Supér, IV. Sér. 21, No. 2, 225-258 (1988; Zbl 0655.42013)] to deduce \(L^p\)-boundedness of the double layer potential. In Chapter 3 applications are given to the solutions of the boundary value problem.
Reviewer: L.Rodino (Torino)
