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Partial differential equations (PDEs): Existence and uniqueness, regularity, boundary conditions, linear and non-linear operators, stability, soliton theory, integrable PDEs, conservation laws, qualitative dynamics.

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How to prove this integral inequality in a 2-D region?

Let $\Omega$ be a 2D region. Now we have a partial differential equation system describing the characteristics of the region: \begin{align*} \nabla \cdot (h_0^3 P_0 \nabla P_0) &= 0 \\ \nabla \cdot [ …