Reference number : GC-2017-Q1-02394, amendment number : , agreement type : G, recipient legal name : Xiao, Jie (Memorial University of Newfoundland), recipient country : CA, recipient province : nan, recipient city : Newfoundland and Labrador|Terre-Neuve-et-Labrador, recipient postal code : nan, program name english : Discovery Grants Program - Individual, program name french : Programme de subventions à la découverte - individuelles, program purpose english : 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))&lt;|grad(u)| -1 grad(u),|grad(u)| -1 grad(u)&gt; 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))&lt;|grad(u)| -1 grad(u),|grad(u)| -1 grad(u)&gt;,
 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&lt;p&lt;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., program purpose french : 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))&lt;|grad(u)| -1 grad(u),|grad(u)| -1 grad(u)&gt; 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))&lt;|grad(u)| -1 grad(u),|grad(u)| -1 grad(u)&gt;,
 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&lt;p&lt;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., agreement title english : Convex Geometric Potential Theory, agreement title french : Convex Geometric Potential Theory, agreement number : RGPIN, agreement value : 100000.0, agreement start date : 2017-05-10, agreement end date : nan, description english : nan, description french : nan, expected results english : nan, expected results french : nan, additional information english : Grant or Award spanning more than one fiscal year. (2017-2018 to 2022-2023), additional information french : Subvention ou bourse octroyée s'appliquant à plus d'un exercice financier. (2017-2018 à 2022-2023), owner organization : nserc-crsng, owner organization title : Natural Sciences and Engineering Research Council of Canada | Conseil de recherches en sciences naturelles et en génie du Canada