For each open, bounded and convex domain $ \Omega \subset \mathbb{R}^{D}, $ $ D\geq 2 $, and each real number $ p > 1, $ we denote by $ u_{p} $ the $ p $-torsion function on $ \Omega $, i.e., the solution of the torsional creep problem $ \Delta_{p}u = -1 $ in $ \Omega $, $ u = 0 $ on $ \partial \Omega $, where $ \Delta _{p}u: = \operatorname{div}(\left\vert \nabla u\right\vert ^{p-2}\nabla u) $ is the $ p $-Laplacian. Let $ T_p(\Omega) $ be the $ p $-torsional rigidity on $ \Omega $, defined as $ T_{p}\left(\Omega \right) : = \int_{\Omega }u_{p}dx $. Define $ T\left(p; \Omega \right) : = \left\vert \Omega \right\vert ^{p-1}T_{p}\left(\Omega \right) ^{1-p} $, where $ |\Omega| $ stands for the Lebesgue measure of $ \Omega $. The main purpose of this paper is to compare the values of $ T(p; \Omega) $ for bounded convex domains having different inradii. We prove that for any $ 0 < a < b $ there exists an explicit constant $ \gamma_{D, p}\in[1/D, 1) $, depending only on the dimension $ D $ and the parameter $ p $, such that $ T(p; \Omega_b)\leq T(p; \Omega_a) $, for all $ \Omega_a\in\mathbb{P}^D(a) $, and $ \Omega_b\in\mathbb{P}^D(b) $, if and only if $ \gamma_{D, p}b\geq a $, where $ \mathbb{P}^D(r) $ denotes the family of convex bounded domains in $ \mathbb{R}^D $ of inradius $ r $. In addition, we discuss the asymptotic equality case, the limiting regimes $ p\rightarrow 1^+ $ and $ p\rightarrow \infty $, and the behaviour of the scale-invariant functional on model families such as rectangles, orthotopes, ellipses, and triangles. We also derive a fixed-volume monotonicity consequence for the unnormalised torsional rigidity, parameterized by the inradius.

In this paper we analyze operators $H=\partial_{t}-\sum_{i, j} a_{i j}(x, t) X_{i} X_{j}$, where the $X_{i} \text { 's }$ are Hörmander vector fields generating a Carnot group and $A=\left[a_{i j}\right]$ is a symmetric and uniformly positive-definite matrix whose entries satisfy double Dini continuity, a strictly weaker condition than Hölder continuity. For these operators, we build a fundamental solution and show a two-sided Gaussian estimate for the latter, as well as upper Gaussian estimates for its derivatives up to weight 2. As a consequence, we prove an existence result for the related Cauchy problem, under a Dini-type condition on the source.

In tissue engineering, it is important to conceive and construct artificial bio-mimetic scaffolds that mimic the native extracellular matrix and are able to foster cell migration, which represents a fundamental process in wound healing and tissue regeneration. In order to do that, cubically symmetric and triply periodic porous structures have emerged as promising candidates, for instance for the in vitro reconstruction of cartilage and bones, also due to their tunable mechanical characteristics and highly interconnected porous architectures that resemble trabecular bone hyperboloidal topography. The problem addressed in this article is the identification of the geometrical characteristics of such scaffolds that best favor cell migration into the porous structures to speed-up their re-population. The method is based on the observation that cell nucleus deformations should be avoided, because they can slow down cell migration, while the pore size should remain sufficiently small to favor cell adhesion to the scaffold walls and the generation of traction forces, that are fundamental for more efficient movements of the cells. Mathematically speaking, at the cellular scale, this leads to the calculation of the size of the largest sphere that can traverse the bio-mimetic scaffold, without being stuck, eventually yielding for each studied geometry a specific ratio between the periodic cell size and the nucleus diameter. These microscopic geometrical descriptors could provide inputs for subsequent macroscopic models of cell migration, where effective motility can be related to the architecture of the scaffold.