In the present paper we establish sharp exponential decay estimates for operator and integral kernels of the (not necessarily self-adjoint) operators L = −(∇− ia) T A(∇ − ia) + V . The latter class includes, in particular, the magnetic Schrödinger operator − (∇ − ia) 2 +V and the generalized electric Schrödinger operator −divA∇+ V . Our exponential decay bounds rest on a generalization of the Fefferman-Phong uncertainty principle to the present context and are governed by the Agmon distance associated to the corresponding maximal function. In the presence of a scale-invariant Harnack inequality, for instance, for the generalized electric Schrödinger operator with real coefficients, we establish both lower and upper estimates for fundamental solutions, thus demonstrating sharpness of our results. The only previously known estimates of this type pertain to the classical Schrödinger operator −∆ + V [29] .

We consider the inverse multiphase Stefan problem, where information on the heat flux on the fixed boundary is missing and must be found along with the temperature and free boundaries. Optimal control framework is pursued, where boundary heat flux is the control, and the optimality criteria consist of the minimization of the L 2 -norm declination of the trace of the solution to the Stefan problem from the temperature measurement on the fixed right boundary. The state vector solves multiphase Stefan problem in a weak formulation, which is equivalent to Neumann problem for the quasilinear parabolic PDE with discontinuous coefficient. Full discretization through finite differences is implemented and discrete optimal control problem is introduced. We prove wellposedness in a Sobolev space framework and convergence of discrete optimal control problems to the original problem both with respect to the cost functional and control. Along the way, the convergence of the method of finite differences for the weak solution of the multiphase Stefan problem is proved. The proof is based on achieving a uniform L ∞ bound, and W 1,1 2 -energy estimate for the discrete multiphase Stefan problem.Key words: Inverse multiphase Stefan problem, quasilinear parabolic PDE with discontinuous coefficent, optimal control, Sobolev spaces, method of finite differences, discrete optimal control problem, energy estimate, embedding theorems, weak compactness, convergence in functional, convergence in control.AMS subject classifications: 35R30, 35R35, 35K20, 35Q93, 65M06, 65M12, 65M32, 65N21. * This research is funded by NSF grant #1359074Inverse Multiphase Stefan Problem (IMSP). Find the functions u(x, t), ξ j (t), j = 1, J, and the boundary heat flux g(t) satisfying (1)-(6),(8).Motivation for the IMSP arose from the modeling of bioengineering problems on the laser ablation of biological tissues through a multiphase Stefan problem (1)-(6). Laser ablation creates three phases -solid (skin), fluid (melted skin) and gas (evaporated skin). Free boundaries ξ 1 (t) and ξ 2 (t) are measuring ablation depth separating solid/fluid and fluid/air regions at the moment t. GR±ε J (g), ε > 0. Then lim ε→0 J * (ε) = J * = lim ε→0 J * (−ε).Proof. The proof of this lemma is very similar to the analogous lemma from [1]. If 0 < ε 1 < ε 2 , then

We extend in two directions the notion of perturbations of Carleson type for the Dirichlet problem associated to an elliptic real second-order divergence-form (possibly degenerate, not necessarily symmetric) elliptic operator. First, in addition to the classical perturbations of Carleson type, that we call additive Carleson perturbations, we introduce scalar-multiplicative and antisymmetric Carleson perturbations, which both allow non-trivial differences at the boundary. Second, we consider domains which admit an elliptic PDE in a broad sense: we count as examples the 1-sided NTA (a.k.a. uniform) domains satisfying the capacity density condition, the 1-sided chord-arc domains, the domains with low-dimensional Ahlfors-David regular boundaries, and certain domains with mixed-dimensional boundaries; thus our methods provide a unified perspective on the Carleson perturbation theory of elliptic operators. Our proofs do not introduce sawtooth domains or the extrapolation method. We also present several applications to some Dahlberg-Kenig-Pipher operators, free-boundary problems, and we provide a new characterization of A ∞ A_{\infty } among elliptic measures.

We prove that the L p ′ -solvability of the homogeneous Dirichlet problem for an elliptic operator L = − div A∇ with real and merely bounded coefficients is equivalent to the L p ′ -solvability of the Poisson Dirichlet problem Lw = H − div F, assuming that H and F lie in certain Carleson-type spaces, and that the domain Ω ⊂ R n+1 , n ≥ 2, satisfies the corkscrew condition and has n-Ahlfors regular boundary. The L p ′solvability of the Poisson problem (with an L p ′ estimate on the non-tangential maximal function) is new even when L = −∆ and Ω is the unit ball. In turn, we use this result to show that, in a bounded domain with uniformly n-rectifiable boundary that satisfies the corkscrew condition, L p ′ -solvability of the homogeneous Dirichlet problem for an operator L = − div A∇ satisfying the Dahlberg-Kenig-Pipher condition (of arbitrarily large constant) implies solvability of the L p -regularity problem for the adjoint operator L * = − div A T ∇, where 1/p + 1/p ′ = 1 and A T is the transpose matrix of A. Contents 4.2. The starlike Lipschitz subdomains Ω ± R 31 4.3. The corona decomposition of Ω 36 4.4. The properties of Ω R 38 5. The almost L-elliptic extension 41 6. The Regularity Problem for DKP operators 42 6.1. A n.t. maximal function estimate for the almost L-elliptic extension 44 6.2. Proof of Proposition 6.2 49 Appendix A. Proofs of auxiliary results 54 References 63

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