\(L^p\)-spectral theory of higher-order elliptic differential operators. (English) Zbl 0955.35019

From the introduction: In this review we describe the \(L^p\) spectral theory of selfadjoint operators acting in either \(L^2(\mathbb{R}^N)\) or \(L^2(\Omega)\), where \(\Omega\) is a region in \(\mathbb{R}^N\), and satisfying the following list of conditions. We start with the expression \[ Hf(x):= \sum_{|\alpha|\leq m, |\beta|\leq m}(-1)^{|\alpha|} D^\alpha\{a_{\alpha,\beta}(x) D^\beta f(x)\}, \] where \(a(x)\) is a bounded measurable complex selfadjoint matrix-valued function, \(\alpha\), \(\beta\) are multi-indices, and \(|\alpha|:= \alpha_1+\cdots+ \alpha_N\). We say that \(H\) is homogeneous of order \(2m\) if the only non-zero terms are those with \(|\alpha|= |\beta|= m\). The domain of such an operator need not contain the space \(C^\infty_c\) of smooth functions with compact support in \(\mathbb{R}^N\) or in \(\Omega\), so we define \(H\) using quadratic form techniques. Namely, we define the quadratic form \(Q\) on \(C^\infty_c\) by \[ Q(f):= \int_\Omega\sum_{|\alpha|\leq m,|\beta|\leq m} a_{\alpha,\beta}(x) D^\beta f(x)\overline{D^\alpha f(x)} dx. \] This form is closable, and the domain of the closure is the Sobolev space \(W^{m,2}_0(\Omega)\) under the following conditions. Let \(S\) denote the set of multi-indices \(\alpha\) such that \(|\alpha|= m\), and let \(\widetilde a(x)\) denote the matrix obtained from \(a(x)\) by restricting the indices to \(S\). We assume that there exist constant nonnegative realf symmetric matrices \(\{c_{i,\alpha,\beta}\}_{\alpha,\beta\in S}\) for \(i= 1,2\), such that \(c_1\leq \widetilde a(x)\leq c_2\) in the sense of matrices for all \(x\in\Omega\) and such that the operators \[ (-1)^m \sum_{|\alpha|= |\beta|= m} c_{i,\alpha,\beta} D^{\alpha+\beta} \] are elliptic in the standard sense, that is, \[ \sum_{|\alpha|= |\beta|= m} c_{i,\alpha,\beta} \xi^{\alpha+ \beta}\geq b_i|\xi|^{2m} \] for some \(b_i> 0\) and all \(\xi\in \mathbb{R}^N\).
The term ‘elliptic operator’ below will always refer to the selfadjoint operator \(H\) associated with the closure of a form \(Q\) satisfying all of the above conditions. This method of approach amounts to the choice of what are called (zero) Dirichlet boundary conditions (DBC). In classical terms these would be the conditions \(D^\alpha f(x)= 0\) for all \(x\in\partial\Omega\) and all \(|\alpha|\leq m-1\), but since we base our calculations upon the quadratic form rather than the operator, we do not use this fact. It should be noted that the operator \((\Delta^2)_{\text{DBC}}\) satisfies the boundary conditions \(f= \partial f/\partial n= 0\) on \(\partial\Omega\), and has quite different spectrum from \((\Delta_{\text{DBC}})^2\).
Our goal is to review results obtained within the last fifteen years on the spectral theory of such operators. We concentrate on those matters which do not naturally fall within the scope of the theory of pseudodifferential operators by avoiding regularity conditions on the coefficients and boundaries except to the extent that it has been discovered that these are really necessary for the validity of theorems of interest. It will be clear that many of the results which we describe are not in a final state. Nevertheless, much is known about the overall shape of the subject, and it seems desirable to make this available to non-experts. We focus on those matters which relate to \(L^p\) properties of the operators, and of the semigroups which those operators generate.


35J30 Higher-order elliptic equations
35P05 General topics in linear spectral theory for PDEs
35-02 Research exposition (monographs, survey articles) pertaining to partial differential equations
35K25 Higher-order parabolic equations
35R05 PDEs with low regular coefficients and/or low regular data
35J10 Schrödinger operator, Schrödinger equation
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