Regularity of \(p\)-harmonic functions on the plane (Q1814010)

In this note we determine the optimal regularity of \(p\)-harmonic functions defined on a plane domain for each exponent \(p\), \(1<p<\infty\). To state the results we use the following function spaces. Theorem 1. The Hölder space \(C^{k,\alpha}_{\text{loc}}(\Omega)\), \(k=0,1,\dots\), \(0<\alpha\leq 1\), is the space of complex-valued functions \(u\in C^ k(\Omega)\) whose \(k\)-th order partial derivatives \(D^ \nu u\), \(|\nu|=k\), are locally Hölder continuous with exponent \(\alpha\), \(C^{k,\alpha}_{\text{loc}} (\Omega)\) is a locally convex linear space with topology determined by the seminorms \[ \| u\|_{C^{k,\alpha} (F)}= \sup_{x\in F} | u(x)|+ \sup_{x,y\in F} \sum_{|\nu| =k} {{| D^ \nu u(x)- D^ \nu u(y)|} \over {| x-y|^ \alpha}}, \] where \(F\) is any compact subset of \(\Omega\). The completion of \(C^ \infty (\Omega)\) in this topology is a proper subspace of \(C^{k,\alpha}_{\text{loc}} (\Omega)\) which we denote by \(C^{k+\alpha}_{\text{loc}} (\Omega)\). For \(\alpha=1\) the space \(C^{k+\alpha}_{\text{loc}} (\Omega)\) coincides with \(C^{k+1} (\Omega)\) whereas functions in \(C^{k+\alpha}_{\text{loc}} (\Omega)\) with \(0<\alpha<1\) are characterized by the condition \[ \sum_{|\nu| =k} | D^ \nu u(x)- D^ \nu u(y)|= o(| x-y|^ \alpha) \] uniformly on compact subsets of \(\Omega\times \Omega\). We denote by \(W^{k,s}_{\text{loc}} (\Omega)\), \(k=1,2,\dots\), \(1\leq s\leq\infty\), the space of functions \(u: \Omega\to \mathbb{C}\) whose distributional derivatives \(D^ \nu u\), \(|\nu| \leq k\), belong to \(L^ s_{\text{loc}} (\Omega)\). Our main result is the following: Theorem. Let \(u\in W^{1,p}_{\text{loc}} (\Omega)\), \(1<p<\infty\), be a \(p\)-harmonic function defined on a plane domain \(\Omega\subset \mathbb{R}^ 2\). Then \[ u\in C^{k,\alpha}_{\text{loc}} (\Omega)\cap W^{k+2}_{\text{loc}} (\Omega), \tag{1} \] where the integer \(k\geq 1\) and the exponent \(\alpha\in (0,1]\) are determined by the equation \[ k+\alpha= {1\over 6} \Biggl( 7+ {1\over {p-1}}+ \sqrt{1+ {{14} \over {p-1}} + {1\over {(p-1)^ 2}}} \Biggr). \] The integrability exponent \(q\) is any number such that \[ 1\leq q< {2\over {2-\alpha}}\leq 2. \] For \(p\neq 2\) the regularity class in (1) is optimal. More precisely, for each \(1<p<\infty\), \(p\neq 2\), there is a \(p\)-harmonic function \(v\in W^{1,p}_{\text{loc}} (\Omega)\) which is not in the class \(C^{k+\alpha}_{\text{loc}}(\Omega)\cup W_{\text{loc}}^{k+2,{2\over {2-\alpha}}} (\Omega)\). The proof of the Theorem substantially exploits and extends the ideas from \textit{B. Bojarski} and the first author, Partial differential equations, Banach Cent. Publ. 19, 25-38 (1987; Zbl 0659.35035)]. A key is the hodograph transformation that converts the \(p\)-harmonic equation onto a linear first order elliptic system. We solve this system by using Fourier series method. A careful examination of the Fourier expansion formula for the solution of the system leads us to the regularity statement. This formula provides non-trivial examples of \(p\)-harmonic functions. Among them there is one showing that our regularity result is the best possible.