The Green function estimates for strongly elliptic systems of second order
By S. Hofmann, Seick Kim. 2007
152 citations recorded by the source corpus.
We establish existence and pointwise estimates of fundamental solutions and Green’s matrices for divergence form, second order strongly elliptic systems in a domain $$\Omega \subseteq {\mathbb{R}}^n, n \geq 3$$ , under the assumption that solutions of the system satisfy De Giorgi-Nash type local Hölder continuity estimates. In particular, our results apply to perturbations of diagonal systems, and thus especially to complex perturbations of a single real equation.
Subfields: Applied Mathematics, Computational Theory and Mathematics
View Paper: https://www.semanticscholar.org/paper/91c00f5a3cf5805e1173f79df170652d84d17427
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In Applied Mathematics
In Computational Theory and Mathematics
Citations in the corpus
- Homogenization of elliptic systems with Neumann boundary conditions (2010, background): “where C depends only on d , m , μ , λ , and τ (see e.g. (cid:8) (cid:9) [17]; the size estimate (5.1) also follows from [13]).”
- The method of layer potentials in Lp and endpoint spaces for elliptic operators with L∞ coefficients (2013, unknown): “In the presence of DG/N/M estimates for L and L∗, by [27], both L and L∗ have fundamental solutions E : {(X,Y ) ∈ R × R : X = Y } → C and E∗(X,Y ) := E(Y,X), respectively, satisfying E(X, ·), E(·,X) ∈ L(2)1,loc(R \ {X}) and Lx,tE(x, t, y, s) = δ(y,s), Ly,sE ∗(y, s, x, t) = Ly,sE(x, t, y, s) = δ(x,t)”
- Green's matrices of second order elliptic systems with measurable coefficients in two dimensional domains (2007, background): “e strategy of Fuchs and showed the existence and pointwise estimate for Green’s matrix in bounded Lipschitz domains Ω ⊂ R2 without imposing any regularity on the coefficients. Recently, Hofmann and Kim [10] gave a unified approach in studying Green ’s functions/matrices in arbitrary domains Ω ⊂ Rn (n ≥”
- Analyticity of layer potentials and $L^{2}$ solvability of boundary value problems for divergence form elliptic equations with complex $L^{\infty}$ coefficients (2007, background): “3See [ HK2 ] for a construction of the fundamental solution.”
- Homogenization and boundary layers (2012, unknown): “Following [16, 12], the Green matrix G(y, ỹ) is defined as the unique matrix function satisfying: a) G is continous over (Ωa × Ωa) \ {y = ỹ}.”
- Uniqueness and Lipschitz stability for the identification of Lam\'e parameters from boundary measurements (2013, background): “ in the case of strongly elliptic systems with L∞ coefficients in dimension n≥ 3, existence of the fundamental solution and of the Green’s function cannot be inferred without additional assumptions. In [HK] Hofmann and Kim prove their existence under the additional information that weak solutions of t”