Nonlinear Potential Theory for elliptic equations Abubakar Mwasa Department of Mathematics, Link¨ oping University Department of Mathematics, Busitema University First Network Meeting for Sida- and ISP-funded PhD Students in Mathematics Stockholm 7–8 March 2017 1 / 7
My Advisors Jana Bj¨ orn Anders Bj¨ orn V. Ssembatya G.I Mirumbe Main advisor Assistant advisor Assistant advisor Assistant advisor Link¨ oping Link¨ oping Makerere Makerere 2 / 7
Nonlinear elliptic equations in unbounded domains via inversions Minimisation of energy using pde e.g. Laplace equation ∆ u = 0 which is equivalent to minimising energy integral Ω |∇ u | 2 dx . � Cousin form: Nonlinear ( p − Laplace) equation for nonlinear situations ∆ p u := div ( |∇ u | p − 2 ∇ u ) = 0; 1 < p < ∞ Problem: Consider a mixed BVP for p − Laplace equation in an open half infinite cylinder in R n . 3 / 7
Schematic representation of the intended problem 4 / 7
Tools and Expectations Main aim: Establish the behaviour of weak solutions at ∞ . Apply tools from pdes on bounded domains to obtain results. Generalise [1] to the nonlinear situation and will contribute to the understanding of boundary regularity on infinite domains Gradually move towards potential theory and pdes on metric spaces. Acquire and extend the research skills in the field of potential theory to my home university. 5 / 7
Reference 1 Bj¨ orn, A.: Regularity at infinity for a mixed problem for degenerate elliptic operators in a half cylinder. Math. Scand. 81 , 101-126(1997) 6 / 7
Tack s˚ a mycket! Thank you! 7 / 7
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