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Regularity theory
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<p>Regularity is a topic of the mathematical study of <a href="/facts/Partial_differential_equation/DyPsBlv7">partial differential equations</a> (PDE) such as <a href="/facts/Laplace%2527s_equation/8n1YkmFS">Laplace's equation</a>, about the <a href="/facts/Integral/V7lyV997">integrability</a> and <a href="/facts/Differentiable/jQqTgLk1">differentiability</a> of <a href="/facts/Weak_solution/udTTFPIX">weak solutions</a>. <a href="/facts/Hilbert%2527s_nineteenth_problem/tDSyXbv1">Hilbert's nineteenth problem</a> was concerned with this concept. </p><p>The motivation for this study is as follows. It is often difficult to construct a classical solution satisfying the PDE in regular sense, so we search for a weak solution at first, and then find out whether the weak solution is smooth enough to be qualified as a classical solution. </p><p>Several theorems have been proposed for different types of PDEs. </p>