| Abstract: |
| In this work, we develop an $L^p$-theory for the existence of weak and strong solutions to the elliptic problem
$$-\text{div\,}(a\nabla u)=f,$$
in a bounded open domain of $\mathbb R^N$, under Dirichlet or Neumann boundary conditions, and under suitable assumptions on the data and on the regularity of the domain boundary.
We also develop an $\boldsymbol L^p$-theory for the existence of weak and strong solutions to the vector-valued problem defined in a bounded open multiply connectde domain of $\mathbb R^3$,
$$\mathbf{curl}(\alpha \mathbf{curl} \boldsymbol h)=\boldsymbol f,$$
considering both perfectly permeable and perfectly conducting boundary conditions, and different assumptions on $\boldsymbol f$ and on the regularity of $\partial\Omega$.
As an application, we study a coupled electromagnetic induction heating problem, obtaining improved regularity results for the solutions compared to those available in the existing literature. |
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