Elliptic Differential-Operator Problems with the Spectral Parameter in Both the Equation and Boundary Conditions and the Corresponding Abstract Parabolic Initial Boundary Value Problems

Abstract

We consider, in UMD Banach spaces, boundary value problems for second order elliptic differential-operator equations with the spectral parameter and boundary conditions containing the parameter in the same order as the equation. An isomorphism and the corresponding estimate of the solution (with respect to the space variable and the parameter) are obtained. Then, an application of the obtained abstract results is given to boundary value problems for second order elliptic differential equations with the parameter in non-smooth domains. Further, the corresponding abstract parabolic initial boundary value problem is treated and an application to initial boundary value problems with time differentiation in boundary conditions is demonstrated.

AMS Subject Classifications: 34G10, 35J25, 47E05, 47D06.

Dedicated to the memory of Alfredo Lorenzi

Appendix

Theorem 20.10

([16, Theorem 5.4.2/1]) Let an operator A be closed, densely defined and invertible in the complex Banach space E and $\|R(\lambda,A)\|_{B(E)} \leq L(1 + \vert \lambda \vert )^{-1}$, for $\vert \arg \lambda \vert \geq \pi -\varphi$, for some $\varphi \in (0,\pi )$. Moreover, let m be a positive integer, p ∈ (1,∞), and $\alpha \in ( \frac{1} {2p},m + \frac{1} {2p})$.

Then, there exists $C \in \mathbb{R}^{+}\ ($ depending only on L, $\varphi$ , m, α, and p) such that, for every $u \in (E,E(A^{m}))_{ \frac{\alpha }{ m}- \frac{1} {2\mathit{mp}},p}$ and $\vert \arg \lambda \vert \leq \varphi$ ,

$$\displaystyle{\int _{0}^{\infty }\|(A +\lambda I)^{\alpha }e^{-x(A+\lambda I)^{\frac{1} {2} } }u\|_{E}^{p}\mathit{dx} \leq C\Big(\|u\|_{ (E,E(A^{m}))_{ \frac{\alpha } {m}- \frac{1} {2\mathit{mp}},p}}^{p} + \vert \lambda \vert ^{p\alpha -\frac{1} {2} }\|u\|_{E}^{p}\Big).}$$