Let D be an open subset of \({\mathbb {R}}^n\) with finite measure, and let \(x_0 \in D\). We introduce the p-Gauss gap of D w.r.t. \(x_0\) to measure how far are the averages over D of the harmonic functions \( u \in L^p(D)\) from \(u(x_0)\). We estimate from below this gap in terms of the ball gap of D w.r.t. \(x_0\), i.e., the normalized Lebesgue measure of \(D \setminus B\), being B the biggest ball centered at \(x_0\) contained in D. From these stability estimates of the mean value formula for harmonic functions in \(L^p\)-spaces, we straightforwardly obtain rigidity properties of the Euclidean balls. We also prove a continuity result of the p-Gauss gap in the Sobolev space \(W^{1,p'}\), where \(p'\) is the conjugate exponent of p.

设D为具有有限度量的\（{\ mathbb {R}} ^ n \）的开放子集，并设\（x_0 \ in D \）。我们引入p -高斯缺口的d WRT \（X_0 \）来衡量多远超过平均值d的调和函数\（U \在L ^ P（d）\）从\（U（X_0）\）。我们从该间隙以下以D wrt \（x_0 \）的球间隙来估算，即\（D \ setminus B \）的归一化Lebesgue度量，它是B以\（x_0 \）为中心的最大球包含在D中。从这些对\（L ^ p \）-空间中谐波函数的平均值公式的稳定性估计中，我们可以直接获得欧几里得球的刚度性质。我们还证明了Sobolev空间\（W ^ {1，p'} \）中p -Gauss间隙的连续性结果，其中\（p'\）是p的共轭指数。