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Elliptic problems and holomorphic functions in Banach spaces. (English) Zbl 1447.35133

The authors consider an open subset \(\Omega \) of \(\mathbb{C}\) (or \(\mathbb{R}^{d}\)), a complex (real) Banach space \(X\) and a function \(f:\Omega\rightarrow X\). They recall that \(f\) is holomorphic (harmonic) if it is complex differentiable (twice partially differentiable with \(\Delta f=0\)). They say that \(f\) is very weakly holomorphic (very weakly harmonic) if there exists a separating subset \(W\subset X^{\prime }\) such that \(x^{\prime}\circ f\) is holomorphic (harmonic) for all \(x^{\prime }\in W\). The first main result of the paper proves that a function \(f:\Omega \rightarrow X\) is holomorphic (harmonic) if and only if it is locally \(L^{1}\)-bounded and very weakly holomorphic (very weakly harmonic). The second main result proves that every function \(f\in L_{\mathrm{loc}}^{1}(\Omega ,X)\) such that \(\Delta f=0\) in a distributional sense has a harmonic representative that is, there exists \(f^{\ast }\in C^{\infty }(\Omega ,X)\) such that \(\Delta f^{\ast }=0\) and \(f^{\ast }=f\) almost everywhere. The authors first prove these results assuming that \(X\) is separable. Considering here the case where \(f\) is very weakly harmonic, they prove that \(f\) is measurable using the Krein-Šmulyan theorem. They then introduce a mollifier \(\rho _{r}\) supported in a ball \(B(0,r)\) and they prove that \(\rho _{r}\ast f=f\) almost everywhere in \(\Omega _{r}=\{\xi \in \Omega ,\mathrm{dist}(\xi ,\partial \Omega )>r\}\) and \(\left\langle \rho _{r}\ast f,x^{\prime }\right\rangle =\rho _{r}\ast\left\langle f,x^{\prime }\right\rangle =\left\langle f,x^{\prime}\right\rangle \) in \(\Omega _{r}\). In the very holomorphic case, the authors introduce the function \(u(z)=\frac{1}{2\pi i}\int_{\left\vert w-z_{0}\right\vert =r}\frac{f(w)}{z-w}dw\), for \(z_{0}\in \Omega \) and \(z\in B(z_{0},r)\), which is proved to be holomorphic and to satisfy \(\left\langle u(z),x^{\prime }\right\rangle =\left\langle f,x^{\prime }\right\rangle \) in \(B(z_{0},r)\). For the second result, the authors also introduce a mollifier and they use Weyl’s lemma. They prove that every function\(f\in L_{\mathrm{loc}}^{1}(\Omega ,X)\) which satisfies the Cauchy-Riemann equations very weakly distributionally has a holomorphic representative. Assuming now that the Banach space \(X\) is not necessarily separable, the authors prove that the space \(Y=\{x^{\prime }\in X^{\prime },x^{\prime }\circ f\) is holomorphic (harmonic)\(\}\) weak-\(\ast \) dense in \(X^{\prime }\) and closed in the weak-\(\ast \) topology, again using the Krein-Šmulyan theorem. In the second part of their paper, the authors analyze some elliptic problems. They prove regularity results of Newtonian potentials and they analyze the Laplace operator \(\Delta _{p}\) with maximal domain in \(L^{p}(\mathbb{R}^{d},X)\), \(1 < p<\infty \). They prove that if \(X\) has the UMD property if and only if \(D(\Delta _{p})=W^{2,p}(\mathbb{R}^{d},X)\). In the last section, the authors consider general linear elliptic operators \(L=a_{ij}D_{ij}+b_{i}D_{i}+c\), with \(a_{ij},b_{i},c\in L^{\infty }(\Omega , \mathbb{R})\), \(a_{ij}\) satisfying symmetry and coercivity properties. They consider the elliptic problem \(Lu=f\) in \(\Omega \) with the boundary condition \(u-\varphi \in W_{0}^{1,p}\) for \(f\in L^{p}(\Omega ,X)\) and \(\varphi \in W^{2,p}(\Omega ,X)\). The last main result of the paper proves that if \(\Omega \) has a \(C^{1,1}\) boundary, \(a\in C(\overline{\Omega }, \mathbb{R}^{d\times d})\), \(c\leq 0\) and \(X\) has the UMD property, then the above elliptic problem has a unique solution. For the proof, the authors consider approximations of a general \(f\in L^{p}(\Omega ,X)\) as sequences of finite sums of tensors.

MSC:

35J25 Boundary value problems for second-order elliptic equations
46B20 Geometry and structure of normed linear spaces
31C05 Harmonic, subharmonic, superharmonic functions on other spaces
32K12 Holomorphic maps with infinite-dimensional arguments or values
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References:

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