Grants and Contributions:

Grant or Award spanning more than one fiscal year. (2017-2018 to 2022-2023)

Originally, potential theory comes from the physical problem of reconstructing a repartition of electric charges inside a body, given a measuring of the electrical field created on the boundary of this body. In terms of analysis this amounts to expressing the values of a function inside a set given the values of the function on the boundary of the set. Recall that current flux F is proportional to difference grad(u)=the gradient of u in electric potential u whenever conductivity is constant. Thus, in the simplest case of a set S without electric charges the problem can be formulated as that of finding a solution u to div(grad(u))=0 in S subject to u's values prescribed by a function f on the boundary of S. But, in reality there exist more complicated forms than div(grad(u)) - one situation is of power-law where F =|grad(u)| p-2 grad(u), leading to the p-Laplace equation div(|grad(u)| p-2 grad (u))=0 in S subject to u=f on the boundary of S - this has been observed in certain materials near the temperatures where the material becomes super-conductive for which p acts as a function of temperature.

Nevertheless, the importance of potential theory over p-Laplacian lies in the study of p-harmonic functions and its links to many areas - in fact - it nicely fills up a position at the interaction of operator theory, complex variables, partial differential equations, topology, probability and geometry. Therefore, potential theory has contributed to and received stimulus from these areas, in its developments.

Interestingly, one-Laplacian (n-1) -1 div(|grad(u)| -1 grad(u)) measures the mean curvature of the level set at each point and infinity-Laplacian (grad 2 (u))<|grad(u)| -1 grad(u),|grad(u)| -1 grad(u)> represents the second derivative in the direction of steepest ascent. From
p -1 |grad(u)| 2-p div(|grad(u)| p-2 grad(u))=p -1 |grad(u)|div(|grad(u)| -1 grad(u))+(1-p -1 )(grad 2 (u))<|grad(u)| -1 grad(u),|grad(u)| -1 grad(u)>,
we see that p-Laplacian may be regarded as a weighted sum of one-Laplacian and infinity-Laplacian. Such an observation leads to an investigation of the convex-geometric-potential-theory (induced by p-Laplacian) that comprises the following five objectives on equilibrium potential and variation capacity.

1. A restriction problem for the Hardy-Morrey-Sobolev space of Riesz potentials of Hardy-Morrey functions.
2. A Minkowski/Yau type minimum/maximum problem for the 1<p<n capacity.
3. An exterior boundary value problem for the p-Laplace equation.
4. A problem for the interplay between the quasilinear diffusion and shape of domain.
5. A valuation-type problem for the p-capacity and its dyadic variant.

The novelty of this DGP-I proposal is to utilize new techniques and methods discovered in differential/Riemannian geometry, probability theory and the p-heat equation with fractional order to treat several basic problems in potential theory and their convex geometric consequences.