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Elliptic boundary problems and the Boutet de Monvel calculus in Besov and Triebel-Lizorkin spaces. (English) Zbl 0873.35023
The subject of this paper are the properties of elliptic boundary problems in a variety of function spaces related to $$L_p$$. To obtain a unified treatment of these questions, Boutet de Monvel’s calculus of pseudo-differential boundary operators is generalized to the full scales of Besov and Triebel-Lizorkin spaces.
In more detail, the calculus is generalized to the Besov spaces $$B^s_{p,q}$$ and Triebel-Lizorkin spaces $$F^s_{p,q}$$ with smoothness-index $$s\in\mathbb{R}$$, integral-exponent $$p\in]0,\infty]$$, and sum-exponent $$q\in]0,\infty]$$ (though only with $$p<\infty$$ for the $$F$$-spaces). These spaces are known to contain for example the Lebesgue, Sobolev, Bessel potential, local Hardy, Slobodetskij, and Hölder-Zygmund spaces as special cases. The continuity properties extend the ones known earlier for either differential problems or, in case of the solution operators (or other strictly pseudo-differential elements of the calculus), for subscales. The results are sharp in terms of the class of the operators and the parameters $$s$$ and $$p$$, showing exactly on which spaces a given operator is defined. On the half-space the operators of class $$-\infty$$ are shown to coincide with those that are continuous from $${\mathcal S}'$$. In various ways, this extends earlier works of Agmon-Douglis-Nirenberg, Solonnikov, Triebel, Franke, Grubb, Grubb-Kokholm, and Chang-Krantz-Stein.
The Fredholm properties of (one- or two-sided) elliptic Green operators are also extended to the full scales: the null-space is an $$(s,p,q)$$-independent $$C^\infty$$-subspace, and a $$C^\infty$$ range-complement may be taken independently of $$(s,p,q)$$. In addition to this, any smooth space is shown to be a complement for all admissible $$(s,p,q)$$ if it so for one such triplet; and this is the case if only it annihilates the range for some $$(s,p,q)$$, which is useful for the treatment of specific problems. As initiated by Grubb-Kokholm, the treated symbol classes are the $$x$$-uniformly estimated ones. To achieve consistency among spaces without embeddings, the trace and singular Green operators of class 0 are in general defined as $$T=K^*e^+$$ and $$G=r^+G_1^*e^+$$ on the half-space; hereby the symbol-kernels of $$K$$ and $$G_1$$ are obtained by taking adjoints of those of $$T$$ and $$G$$, respectively. Earlier results of Buy Huy Qui and Päivärinta for pseudo-differential operators of class $$S^d_{1,0}$$ on the Euclidean space is also a main ingredient.
The results are established on smooth bounded sets in $$\mathbb{R}^n$$, for injectively, surjectively, or two-sidedly elliptic operators; possibly of multi-order and multi-class in the Douglis-Nirenberg sense and possibly acting in vector bundles. Applications to the decomposition of vector fields into divergence-free and gradient fields are indicated together with solvability and regularity results for semilinear problems such as the stationary Navier-Stokes equations.

##### MSC:
 35J15 Second-order elliptic equations 47G30 Pseudodifferential operators 46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems 35S05 Pseudodifferential operators as generalizations of partial differential operators
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