Diffusion equations and geometric inequalities
Given a bounded continuous function \(c: R^{n} \to R\), a positive parameter \(\sigma\) and a point \(x \in R^{n}\), the diffusion equation \[ {{\partial v}\over{\partial t}} = {{\sigma^2}\over{2}} \Delta v - {{1}\over{\sigma^2}} c(x) v \] has a fundamental solution. We denote this solution by \(p_{\sigma,c}(t,x,y)\). The main purpose of this paper is to study the relation between the solutions \(p_{\sigma,c}\) with different parameters \(\sigma\) and \(c\). Explicitly, the author proves that for given parameters \((\sigma_0, c_0, x_0)\) and \((\sigma_1, c_1, x_1)\), if a function \(c_{\theta}\) satisfies \[ {1\over\sigma_{\theta}} c_{\theta}(x_{\theta}) \leq {\theta_0\over\sigma_0} c_0(x_0) + {\theta_1\over\sigma_1} c_1(x_1), \] where \(\theta_0, \theta_1 > 0\), \(\sigma_{\theta} = \theta_0 \sigma_0 + \theta_1 \sigma_1\) and \(x_{\theta} = \theta_0 x_0 + \theta_1 x_1\), then the corresponding solutions to the above diffusion equations satisfy the following geometric inequality \[ (\sigma_{\theta}^n p_{\sigma_{\theta},c_{\theta}}(t,x_{\theta},y_{\theta}))^{\sigma_{\theta}} \geq (\sigma_0^n p_{\sigma_0,c_0}(t,x_0,y_0))^{\theta_0 \sigma_0} (\sigma_1^n p_{\sigma_1,c_1}(t,x_1,y_1))^{\theta_1 \sigma_1}. \]
scientific article; zbMATH DE number 1436913