Local $W^{1,p}$ solutions of real-linear Cauchy–Riemann equations with prescribed value

JHolomorphic.exists_local_weak_solution

Data and hypotheses. Let $n \in \mathbb{N}$ (including $n = 0$, in which case $\mathbb{C}^0 = \{0\}$ and every statement below about $\mathbb{C}^n$-valued maps is trivially satisfiable). Write $\mathbb{C}^n$ for the Euclidean (Hermitian) space $\mathbb{C}^n$ with its standard norm, its complex scalar multiplication, and its underlying real vector space structure. Let $p \in \mathbb{R}$ with

Let

be any map assigning to each $z \in \mathbb{C}$ a continuous real-linear operator $A(z)$ on $\mathbb{C}^n$ (no complex linearity, continuity or smoothness in $z$ is assumed). The only hypothesis on $A$ is that it lies in $L^p$ on the open unit disc $D_1 = \{z \in \mathbb{C} : |z| < 1\}$ with respect to Lebesgue measure, using the operator norm on $\mathcal{L}_{\mathbb{R}}(\mathbb{C}^n,\mathbb{C}^n)$: that is, $A$ is almost-everywhere strongly measurable on $D_1$ and

Nothing is assumed about $A$ outside $D_1$. Finally, let $u_0 \in \mathbb{C}^n$ be an arbitrary vector.

Conclusion. For all sufficiently small $\varepsilon > 0$ — precisely: there exists $\delta > 0$ such that for every $\varepsilon$ with $0 < \varepsilon < \delta$ — the following holds, with $D_\varepsilon = \{z \in \mathbb{C} : |z| < \varepsilon\}$ the open disc of radius $\varepsilon$ about $0$ and all measures being Lebesgue measure on $\mathbb{C} \cong \mathbb{R}^2$. There exist maps $u, u_s, u_t : \mathbb{C} \to \mathbb{C}^n$ (which may depend on $\varepsilon$; their values outside $D_\varepsilon$ are unconstrained) such that:

1. $u$ is continuous on $D_\varepsilon$ (as a map restricted to $D_\varepsilon$);
2. $u(0) = u_0$;
3. $u_s$ and $u_t$ are weak partial derivatives of $u$ on $D_\varepsilon$ in the following sense: for every real-valued test function $\varphi : \mathbb{C} \to \mathbb{R}$ that is $C^\infty$ (as a function of two real variables), has compact support, and whose closed support $\overline{\{\varphi \ne 0\}}$ is contained in the open disc $D_\varepsilon$, both identities

hold, where $z = s + i t$, $\partial_s \varphi(z)$ is the real Fréchet derivative of $\varphi$ at $z$ applied to the vector $1$, and $\partial_t \varphi(z)$ is that derivative applied to the vector $i$. The integrals are Bochner integrals of $\mathbb{C}^n$-valued functions over all of $\mathbb{C}$, with the convention that the integral of a non-integrable function is defined to be $0$ (so each identity is an equation between these conventional values); 4. $u$, $u_s$ and $u_t$ each lie in $L^p(D_\varepsilon)$: each is almost-everywhere strongly measurable on $D_\varepsilon$ and has finite integral of the $p$-th power of its norm over $D_\varepsilon$; 5. for Lebesgue-almost every $z \in D_\varepsilon$,

where $i \cdot u_t(z)$ is complex scalar multiplication by $i$ in $\mathbb{C}^n$ and $A(z)(u(z))$ is the real-linear operator $A(z)$ applied to the vector $u(z)$.

The exponent $p$ enters only through the $L^p$ conditions (the hypothesis on $A$ over $D_1$ and conclusion 4 over $D_\varepsilon$); no Sobolev space, regularity beyond continuity, or uniqueness of $u$ is asserted, and no relation between $\delta$ and $A$, $u_0$, $n$ or $p$ is stated.