Let \(\Gamma\) be a closed smooth Jordan curve in the complex plane \(\mathbb{C}\),G be a bounded domain in \(\mathbb{C}\) with the boundary \(\Gamma\), and let \(\overline{G}=G\cup\Gamma\). We study functions that are continuous in \(\mathbb{C}\setminus G\) and harmonic in \(\mathbb{C}\setminus\overline{G}\) that grow more slowly than the function \(|z|^2\) at \(z\to\infty\). It is shown that, if in the class of such functions there exists a solution to the overdetermined Neumann boundary value problem in which the function equals zero on \(\Gamma\) and \(\mu\) -almost everywhere on \(\Gamma\) there exists and equals one the normal derivative of this function, then the domain G is a disk (Theorem 1). In this case, the solution is unique and it coincides up to a constant with the fundamental solution for the Laplace operator in \(\mathbb{C}\) with a singularity in the center of the disk G. The proof of Theorem 1 is based on the application of the conformal mapping of the exterior of the unit disk onto the domain \(\mathbb{C}\setminus \overline{G}\). This mapping allows us to reduce the original problem for the domain \(\mathbb{C}\setminus \overline{G}\) to overdetermined boundary value problem for the exterior of the disk in which the main difficulty is the heterogeneity of the boundary condition for the normal derivative. To study this condition some subtle results were required on the boundary properties of a function that performs the indicated conformal mapping as well as some properties of the Hardy classes H_p in the unit disk. Theorem 2 of the paper shows that in the general case the conditions in Theorem 1 cannot be relaxed. It states the existence of a bounded domain \(G\subset\mathbb{C}\) different from a disk with a smooth Jordan boundary \(\Gamma\) and functions \(f_1,f_2,f_3\in C(\mathbb{C}\setminus G)\) harmonic in \(\mathbb{C}\setminus\overline{G}\) for each of which exactly one of the conditions of Theorem 1 is not satisfied.

令\（\ Gamma \）为复平面\（\ mathbb {C} \）中的闭合平滑约旦曲线，G为\（\ mathbb {C} \）中边界为\（\ Gamma \ ），然后让\（\ overline {G} = G \ cup \ Gamma \）。我们研究\（\ mathbb {C} \ setminus G \）中连续的函数和\（\ mathbb {C} \ setminus \ overline {G} \）中的谐波 比函数\（| z | ^ 2 \）在\（z \ to \ infty \）处。结果表明，如果在此类函数的类别中，存在超定Neumann边值问题的解决方案，其中该函数在\（\ Gamma \）上等于零，\（\ mu \）  -在\（\ Gamma \）上几乎到处都存在并且等于该函数的正态导数，则域G是一个磁盘（定理1）。在这种情况下，解决方案是唯一的，并且与\（\ mathbb {C} \）中的Laplace运算符的基本解决方案一致，并且在磁盘G的中心具有奇异性。定理1的证明是基于单位磁盘外部到域\（\ mathbb {C} \ setminus \ overline {G} \）的共形映射的应用。此映射使我们可以减少域\（\ mathbb {C} \ setminus \ overline {G} \）的原始问题解决了磁盘外缘的过高边界值问题，其中主要困难是正态导数边界条件的不均匀性。为了研究这种情况，需要对执行指示的共形映射的函数的边界属性以及单位圆图中的Hardy类H _p的某些属性进行微妙的研究。本文的定理2表明，在一般情况下，定理1的条件不能放宽。它指出存在有界域\（G \ subset \ mathbb {C} \）与具有光滑Jordan边界\（\ Gamma \）和函数C（\ mathbb的函数\（f_1，f_2，f_3 \ {C} \ setminus G）\）谐波输入\（\ mathbb {C} \ setminus \ overline {G} \），其中每个定理1的条件之一都不满足。