# Physics interpretation of Sobolev space

What is the physics interpretation of Sobolev space? $$H^{s,p}:=\left\{u\in L^p(\mathbb{R}^n):\mathcal{F}^{-1}((1+|\cdot|^2)^{s/2}\mathcal{F}(u))\in L^p(\mathbb{R}^n)\right\}$$, $$s\geq 0,\, 1 (Sobolev space) For example, if $$n=3,\,s=2,\,p=2$$. Is there an interpretation?

Ask this, to find out if the equation $$\mathcal{F}^{-1}((1+|\cdot|^2)^{s/2}\mathcal{F}(u))=g$$ with $$g\in L^2(\mathbb{R}^n)$$ has any physical interpretation or application.

• This seems similarly ill-defined to asking "What is the physics interpretation of $L^2(\mathbb{R})$?". There are many parts of physics where square-integrable functions play a role and that space is relevant, but none of these is really "the physics interpretation" of it - without physical context given, that space is just a mathematical construct without any inherent physical meaning. – ACuriousMind Jun 13 at 13:19

The (non-homogeneous) Sobolev spaces $$H^{p,s}$$ are the subspaces of $$L^p$$ of functions that admit $$s$$ weak derivatives belonging to the same Lebesgue space, where a weak derivative is a derivative in the sense of distributions.
The foremost ones of physical significance are the ones $$H^s=H^{2,s}$$ where the base space is $$L^2$$. This is due to the fact that $$L^2$$ is the prototypical Hilbert space for nonrelativistic quantum mechanics.
As a concrete example, $$H^2$$ is the natural domain of definition, and self-adjointness, of the Laplace operator $$-\Delta$$, that is the kinetic energy of a quantum particle.
Another example may be given by nonlinear Schrödinger equations of the type $$i\partial_t \psi = -\Delta \psi + \lvert \psi\rvert^{p-1} \psi \; .$$ Such equations are crucial to describe the effective behavior of condensed matter systems, such as the motion of atoms in a BEC. The existence and uniqueness of solutions to such equations is typically investigated and proved in Sobolev spaces $$H^s$$.
• A distribution possesses a weak derivatives of any order. I think that by "admits" @yuggib means that the $s$-order weak derivatives of elements of $H^{p,s}$ are still elements of $L^p$. Is this correct? – mike stone Jun 13 at 13:02
By stretching the question just a little bit, there is in fact a very physical interpretation. For a Euclidean QFT with say a scalar field $$\phi$$ in $$d$$ spacetime dimensions, the latter is typically (when Nelson-Symanzik positivity holds in addition to Osterwalder-Schrader positivity) realized as a random Schwartz distribution in $$\mathscr{S}'(\mathbb{R}^d)$$. One may then ask for which (weighted inhomogeneous) Besov spaces $$B_{p,q}^{s}$$ is $$\phi$$ almost surely in that space. The typical answer is: for all $$s<-\Delta_{\phi}$$ where $$\Delta_{\phi}$$ is an important physical quantity, namely the scaling dimension of the field $$\phi$$. This is just the distributional generalization of the Kolmogorov-Chentsov Theorem.