## Solving partial differential equations and systems by techniques of harmonic analysis.

Project: Research

Status Finished 1/12/07 → 30/11/10
Total award £357,700.00 EPSRC EP/F014589/1
Period 1/12/07 → 30/11/10

### Key findings

The main objective of the research was study of elliptic partial differential equations. Partial differential equations are used to mathematically describe behaviour of many real life phenomena and arise practically everywhere. Equations of this type can be encountered in physics, material science, geometry, probability and many other disciplines.
In many real life applications, the equations that arise have certain singularities. For example the domain of equation can have corners, cusps or the coefficients of equation itself might be discontinuous. Here the discontinuity of coefficients is the mathematical expression of the fact that many materials contain impurities (foreign objects) that somewhat change the properties of studied objects. For these reasons it is very important to consider these situations mathematically.
Our research have made substrantial progress in studying these phenomena. We have established solvability of the Dirichlet boundary value problem under assumption that the coefficients satisfy certain Carleson condition. We have established this result for both divergence and non-divergence form elliptic equations. We have also studied the Dirichlet problem with BMO data and have established equivalence between solvability of this problem and so called A_\infty condition.

• ## BMO Solvability and the A ∞ Condition for Elliptic Operators

Research output: Contribution to journalArticle

• ## Elliptic equations in the plane satisfying a Carleson measure condition

Research output: Contribution to journalArticle

• ## $L^p$ Solvability of the Stationary Stokes Problem in Domains with Conical Singularity in Any Dimension

Research output: Contribution to journalArticle

• ## The L-p Dirichlet problem for second-order, non-divergence form operators: solvability and perturbation results

Research output: Contribution to journalArticle

• ## The regularity problem for elliptic operators with boundary data in Hardy–Sobolev space HS

Research output: Contribution to journalArticle