Elliptic boundary problems and the Boutet de Monvel calculus in Besov and Triebel-Lizorkin spaces.

*(English)*Zbl 0873.35023The 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.

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.

Reviewer: J.Johnsen (København)

##### 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 |