In mathematics, a Sobolev space is a vector space of functions equipped with a norm that is a combination of Lp-norms of the function together with its derivatives up to a given order. The derivatives are understood in a suitable weak sense to make the space complete, i.e. a Banach space. Intuitively, a Sobolev space is a space of functions possessing sufficiently many derivatives for some application domain, such as partial differential equations, and equipped with a norm that measures both the size and regularity of a function. Sobolev spaces are named after the Russian mathematician Sergei Sobolev. Their importance comes from the fact that weak solutions of some important partial differential equations exist in appropriate Sobolev spaces, even when there are no strong solutions in spaces of continuous functions with the derivatives understood in the classical sense.
Motivation Throughout the article, Ω {\displaystyle \Omega } is an open subset of R n . {\displaystyle \mathbb {R} ^{n}.}
There are many criteria for smoothness of mathematical functions. The most basic criterion may be that of continuity. A stronger notion of smoothness is that of differentiability (because functions that are differentiable are also continuous) and a yet stronger notion of smoothness is that the derivative also be continuous (these functions are said to be of class C 1 {\displaystyle C^{1}} — see Differentiability classes). Differentiable functions are important in many areas, and in particular for differential equations. In the twentieth century, however, it was observed that the space C 1 {\displaystyle C^{1}} (or C 2 {\displaystyle C^{2}} , etc.) was not exactly the right space to study solutions of differential equations. The Sobolev spaces are the modern replacement for these spaces in which to look for solutions of partial differential equations. Quantities or properties of the underlying model of the differential equation are usually expressed in terms of integral norms. A typical example is measuring the energy of a temperature or velocity distribution by an L 2 {\displaystyle L^{2}} -norm. It is therefore important to develop a tool for differentiating Lebesgue space functions. The integration by parts formula yields that for every u ∈ C k ( Ω ) {\displaystyle u\in C^{k}(\Omega )} , where k {\displaystyle k} is a natural number, and for all infinitely differentiable functions with compact support φ ∈ C c ∞ ( Ω ) , {\displaystyle \varphi \in C_{c}^{\infty }(\Omega ),}
∫ Ω u D α φ d x = ( − 1 ) | α | ∫ Ω φ D α u d x , {\displaystyle \int _{\Omega }u\,D^{\alpha \!}\varphi \,dx=(-1)^{|\alpha |}\int _{\Omega }\varphi \,D^{\alpha \!}u\,dx,}
where α = ( α 1 , . . . , α n ) {\displaystyle \alpha =(\alpha _{1},...,\alpha _{n})} is a multi-index of order | α | = k {\displaystyle |\alpha |=k} and we are using the notation:
D α f = ∂ | α | f ∂ x 1 α 1 … ∂ x n α n . {\displaystyle D^{\alpha \!}f={\frac {\partial ^{|\alpha |}\!f}{\partial x_{1}^{\alpha _{1}}\dots \partial x_{n}^{\alpha _{n}}}}.}
The left-hand side of this equation still makes sense if we assume u {\displaystyle u} to be only locally integrable. If there exists a locally integrable function v {\displaystyle v} , such that
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