Sobolev

Sobolev refers to Sergei Lvovich Sobolev, a Russian mathematician (1908–1989) whose work established foundational tools for functional analysis and the theory of partial differential equations. He introduced function spaces designed to control both a function and its derivatives, providing a framework for studying weak solutions to PDEs.

In a domain Ω ⊂ R^n, the Sobolev space W^{k,p}(Ω) consists of functions u in L^p(Ω) whose weak derivatives D^α u of order |α| ≤ k also lie in L^p(Ω). It is equipped with the norm (sum_{|α|≤k} ∥D^α u∥_{L^p}^p)^{1/p}. When p = 2, W^{k,2}(Ω) is a Hilbert space, often denoted H^k(Ω).

Fractional Sobolev spaces W^{s,p}(Ω) generalize the concept to non-integer smoothness s. On R^n, these can be defined via Fourier transforms or through Gagliardo-type semi-norms. The special case H^s = W^{s,2} is widely used in analysis and PDE theory.

Key properties include Sobolev embedding theorems, which describe how W^{k,p}(Ω) or W^{s,p}(Ω) continuously embed into L^q(Ω) or into spaces of continuous functions under certain relations between n, p, k, and q. Compact embeddings (Rellich-Kondrachov theorem) and trace theorems on boundaries are central results with numerous applications.

Applications of Sobolev spaces span existence, uniqueness, and regularity results for elliptic and parabolic PDEs, variational methods, and the study of functional-analytic aspects of PDEs. Generalizations include Sobolev spaces on manifolds, weighted and variable-exponent spaces, and anisotropic or vector-valued variants.