View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by University of Birmingham Research Portal University of Birmingham The method of layer potentials in Lp and endpoint spaces for elliptic operators with L coefficients Hofmann, Steve; Mitrea, Marius; Morris, Andrew DOI: 10.1112/plms/pdv035 License: None: All rights reserved Document Version Peer reviewed version Citation for published version (Harvard): p Hofmann, S, Mitrea, M & Morris, A 2015, 'The method of layer potentials in L and endpoint spaces for elliptic operators with L coefficients', London Mathematical Society. Proceedings , vol. 111, no. 3, pp. 681-716. https://doi.org/10.1112/plms/pdv035 Link to publication on Research at Birmingham portal General rights Unless a licence is specified above, all rights (including copyright and moral rights) in this document are retained by the authors and/or the copyright holders. 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Take down policy While the University of Birmingham exercises care and attention in making items available there are rare occasions when an item has been uploaded in error or has been deemed to be commercially or otherwise sensitive. If you believe that this is the case for this document, please contact [email protected] providing details and we will remove access to the work immediately and investigate. Download date: 01. Mar. 2020 THE METHOD OF LAYER POTENTIALS IN Lp AND ENDPOINT SPACES FOR ELLIPTIC OPERATORS WITH L1 COEFFICIENTS STEVE HOFMANN, MARIUS MITREA, AND ANDREW J. MORRIS Abstract. We consider layer potentials associated to elliptic operators Lu = − div Aru n+1 acting in the upper half-space R+ for n ≥ 2, or more generally, in a Lipschitz graph domain, where the coefficient matrix A is L1 and t-independent, and solutions of Lu = 0 satisfy interior estimates of De Giorgi/Nash/Moser type. A “Calderon-Zygmund”´ the- ory is developed for the boundedness of layer potentials, whereby sharp Lp and endpoint space bounds are deduced from L2 bounds. Appropriate versions of the classical “jump- relation” formulae are also derived. The method of layer potentials is then used to es- tablish well-posedness of boundary value problems for L with data in Lp and endpoint spaces. Contents 1. Introduction1 2. Jump relations and definition of the boundary integrals9 3. A “Calderon-Zygmund”´ Theory for the boundedness of layer potentials: Proof of Theorem 1.2 11 4. Solvability via the method of layer potentials: Proof of Theorem 1.4 25 5. Boundary behavior of solutions 28 6. Appendix: Auxiliary lemmata 33 References 37 1. Introduction Consider a second order, divergence form elliptic operator (1.1) L = −div A(x)r in Rn+1 := fX = (x; t): x 2 Rn; t 2 Rg; where the gradient and divergence act in all n+1 variables, and A is an (n+1)×(n+1) matrix of L1, t-independent, complex coefficients, satisfying the uniform ellipticity condition n+1 −1 2 X (1.2) Λ jξj ≤ Re hA(x) ξ; ξi := Re Ai j(x) ξ j ξi; kAk1 ≤ Λ; i; j=1 Date: June 26, 2014. The work of the first two authors has been supported in part by NSF. The work of the third author has been supported in part by EPSRC grant EP/J010723/1 and Marie Curie IRSES project 318910. 2010 Mathematics Subject Classification. 35J25, 58J32, 31B10, 31B15, 31A10, 45B05, 47G10, 78A30. Key words: elliptic operators, layer potentials, Calderon-Zygmund´ theory, atomic estimates, boundary value problems. 1 THE METHOD OF LAYER POTENTIALS 2 for some Λ 2 (0; 1), for all ξ 2 Cn+1, and for almost every x 2 Rn. The operator L is interpreted in the usual weak sense via the accretive sesquilinear form associated with (1.2). In particular, we say that u is a solution of Lu = 0, or simply Lu = 0, in an R ⊂ Rn+1 2 2 r · r 2 1 open set Ω , if u L1; loc(Ω) and Rn+1 A u Φ = 0 for all Φ C0 (Ω). Throughout the paper, we shall impose the following “ standard assumptions”: (1) The operator L = − div Ar is of the type defined in (1.1) and (1.2) above, with t-independent coefficient matrix A(x; t) = A(x). (2) Solutions of Lu = 0 in open subsets of Rn+1 satisfy the De Giorgi/Nash/Moser (DG/N/M) estimates defined in (1.3) and (1.4) below. The paper has two principal aims. First, we prove sharp Lp and endpoint space bounds for layer potentials associated to any operator L that, along with its Hermitian adjoint L∗ = − div A∗r, where A∗ is the conjugate transpose of the matrix A, satisfies the standard assumptions. These results are of “Calderon-Zygmund”´ type, in the sense that the Lp and endpoint space bounds are deduced from L2 bounds. Second, we use the layer potential method to obtain well-posedness results for boundary value problems for certain classes of such operators L. The precise definitions of the layer potentials, and a brief historical summary of previous work (including the known L2 bounds), is given below. Let us now discuss some preliminaries in order to state the main results of the paper. d For any measurable subset E of R (typically d = n + 1 or d = n), the symbol 1E denotes the characteristic function of E defined on Rd, the symbol jEj denotes the d-dimensional R E f jEj−1 f Lebesgue measure of , and E := E denotes the integral mean of a measurable function f on E. For notational convenience, capital letters will often be used to denote n+1 n points in R , e.g., X = (x; t>); Y = (y; s), where x; y 2 R and t; s 2 R. In this notation, let B(X; r):= fY 2 Rn+1 : jX − Yj < rg and ∆(x; r):= fy 2 Rn : jx − yj < rg denote, respectively, balls of radius r > 0 in Rn+1 and in Rn. The letter Q will be used to denote a generic cube in Rn with sides parallel to the co-ordinate axes and side length `(Q). The convention is adopted whereby C denotes a finite positive constant that may change from one line to the next but depends only on the relevant preceding hypotheses. We will often write Cp to emphasize when such a constant depends on a specific parameter p. We may also write a . b to denote a ≤ Cb, and a ≈ b to indicate that a . b . a, for some a; b 2 R. De Giorgi/Nash/Moser (DG/N/M) estimates. We say that a locally square integrable function u is locally H¨oldercontinuous, or equivalently, satisfies De Giorgi/Nash (DG/N) n+1 estimates in an open set Ω ⊂ R , if there is a positive constant C0 < 1, and an exponent α 2 (0; 1], such that for any ball B = B(X; R) whose concentric double 2B := B(X; 2R) is contained in Ω, we have !α !1=2 jY − Zj 2 (1.3) ju(Y) − u(Z)j ≤ C0 juj ; R 2B Y; Z 2 B u u − u u ? u whenever . Since = B + B , it is clear that any function satisfying (1.3) also satisfies Moser’s local boundedness estimate (see [48]) > > !1=2 2 (1.4) sup ju(Y)j ≤ Ce0 juj : Y2B 2B Moreover, as is well-known, (1.4) self improves? to !1=r r (1.5) sup ju(Y)j ≤ Cr juj ; 8 r 2 (0; 1): Y2B 2B ? THE METHOD OF LAYER POTENTIALS 3 Remark 1.1. It is well-known (see [19, 48, 49]) that when the coefficient matrix A is real, solutions of Lu = 0 satisfy the DG/N/M estimates (1.3) and (1.4), and the rele- vant constants depend quantitatively on ellipticity and dimension only (for this result, the matrix A need not be t-independent). Moreover, estimate (1.3), which implies (1.4), is stable under small complex perturbations of the coefficients in the L1 norm (see, e.g., [23, Chapter VI] or [2]). Therefore, the standard assumption (2) above holds automatically for small complex perturbations of real symmetric elliptic coefficients. We also note that in the t-independent setting considered here, the DG/N/M estimates always hold when the ambient dimension n + 1 is equal to 3 (see [1, Section 11]). We shall refer to the following quantities collectively as the “standard constants”: the dimension n in (1.1), the ellipticity parameter Λ in (1.2), and the constants C0 and α in the DG/N/M estimates (1.3) and (1.4). In the presence of DG/N/M estimates for L and L∗, by [27], both L and L∗ have fun- damental solutions E : f(X; Y) 2 Rn+1 × Rn+1 : X , Yg ! C and E∗(X; Y):= E(Y; X), 2 n+1 respectively, satisfying E(X; ·); E(·; X) 2 L1; loc(R n fXg) and ∗ ∗ ∗ (1.6) Lx;t E(x; t; y; s) = δ(y;s); Ly;s E (y; s; x; t) = Ly;s E(x; t; y; s) = δ(x;t); n+1 where δX denotes the Dirac mass in R at the point X.