## 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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