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PDEs (Definition)

Differential equations that contain unknown multivariable functions and their partial derivatives.
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The statement of the theorem

Let Ω⊂Rn\Omega \subset \mathbb{R}^n be an open domain, and let u:Ω→Ru: \Omega \to \mathbb{R} be the unknown function. A Partial Differential Equation (PDE) is formally defined by an equation of the structure L(u)=f(x)\mathcal{L}(u) = f(x), where L\mathcal{L} is a linear differential operator of order kk, and f(x)f(x) is a known source function (or forcing term).\n\nFormally, the operator L\mathcal{L} is defined as:\nL(u)=∑∣α∣≤kaα(x)Dαu\mathcal{L}(u) = \sum_{|\alpha| \le k} a_{\alpha}(x) D^{\alpha} u \n\nHere:\n1. Dαu=∂∣α∣u∂x1α1⋯∂xnαnD^{\alpha} u = \frac{\partial^{|\alpha|} u}{\partial x_1^{\alpha_1} \cdots \partial x_n^{\alpha_n}} is the α\alpha-th partial derivative, where α=(α1,…,αn)\alpha = (\alpha_1, \dots, \alpha_n) is a multi-index with non-negative integer components αi\alpha_i. The order of the operator is k=max⁡α(∣α∣)k = \max_{\alpha} (|\alpha|) for which aα(x)≢0a_{\alpha}(x) \not\equiv 0.\n2. aα(x):Ω→Ra_{\alpha}(x): \Omega \to \mathbb{R} are the coefficient functions, assumed to be sufficiently smooth (e.g., aα∈Ck(Ω)a_{\alpha} \in C^k(\Omega)).\n3. The PDE is the equation L(u)=f(x)\mathcal{L}(u) = f(x), which seeks a solution u∈Ck(Ω)u \in C^k(\Omega) satisfying the equation pointwise on Ω\Omega. The classification of the PDE (e.g., elliptic, parabolic, hyperbolic) depends on the principal part of the operator L\mathcal{L} (i.e., the coefficients aαa_{\alpha} where ∣α∣=k|\alpha|=k).