In mathematics, and more precisely, in functional Analysis and PDEs, the Schauder estimates are a collection of results due to Juliusz Schauder (1934, 1937) concerning the regularity of solutions to linear, uniformly elliptic partial differential equations. The estimates say that when the equation has appropriately smooth terms and appropriately smooth solutions, then the Hölder norm of the solution can be controlled in terms of the Hölder norms for the coefficient and source terms. Since these estimates assume by hypothesis the existence of a solution, they are called a priori estimates.

There is both an interior result, giving a Hölder condition for the solution in interior domains away from the boundary, and a boundary result, giving the Hölder condition for the solution in the entire domain. The former bound depends only on the spatial dimension, the equation, and the distance to the boundary; the latter depends on the smoothness of the boundary as well.

The Schauder estimates are a necessary precondition to using the method of continuity to prove the existence and regularity of solutions to the Dirichlet problem for elliptic PDEs. This result says that when the coefficients of the equation and the nature of the boundary conditions are sufficiently smooth, there is a smooth classical solution to the PDE.

Notation

The Schauder estimates are given in terms of weighted Hölder norms; the notation will follow that given in the text of D. Gilbarg and Neil Trudinger (1983).

The supremum norm of a continuous function ${\displaystyle f\in C(\Omega )}$ is given by

${\displaystyle |f|_{0;\Omega }=\sup _{x\in \Omega }|f(x)|}$

For a function which is Hölder continuous with exponent ${\displaystyle \alpha }$, that is to say ${\displaystyle f\in C^{0,\alpha }(\Omega )}$, the usual Hölder seminorm is given by

${\displaystyle [f]_{0,\alpha$ ;\Omega }=\sup _{x,y\in \Omega }{\frac {|f(x)-f(y)|}{|x-y|^{\alpha }}}.}

The sum of the two is the full Hölder norm of f

${\displaystyle |f|_{0,\alpha$ ;\Omega }=|f|_{0;\Omega }+[f]_{0,\alpha ;\Omega }=\sup _{x\in \Omega }|f(x)|+\sup _{x,y\in \Omega }{\frac {|f(x)-f(y)|}{|x-y|^{\alpha }}}.}

For differentiable functions u, it is necessary to consider the higher order norms, involving derivatives. The norm in the space of functions with k continuous derivatives, ${\displaystyle C^{k}(\Omega )}$, is given by

${\displaystyle |u|_{k;\Omega }=\sum _{|\beta |\leq k}\sup _{x\in \Omega }|D^{\beta }u(x)|}$

where ${\displaystyle \beta }$ ranges over all multi-indices of appropriate orders. For functions with kth order derivatives which are Holder continuous with exponent ${\displaystyle \alpha }$, the appropriate semi-norm is given by

${\displaystyle [u]_{k,\alpha$ ;\Omega }=\sup _{\stackrel {x,y\in \Omega }{|\beta |=k}}{\frac {|D^{\beta }u(x)-D^{\beta }u(y)|}{|x-y|^{\alpha }}}}

which gives a full norm of

${\displaystyle |u|_{k,\alpha$ ;\Omega }=|u|_{k;\Omega }+[u]_{k,\alpha ;\Omega }=\sum _{|\beta |\leq k}\sup _{x\in \Omega }|D^{\beta }u(x)|+\sup _{\stackrel {x,y\in \Omega }{|\beta |=k}}{\frac {|D^{\beta }u(x)-D^{\beta }u(y)|}{|x-y|^{\alpha }}}.}

For the interior estimates, the norms are weighted by the distance to the boundary

${\displaystyle d_{x}=d(x,\partial \Omega )}$

raised to the same power as the derivative, and the seminorms are weighted by

${\displaystyle d_{x,y}=\min(d_{x},d_{y})}$

raised to the appropriate power. The resulting weighted interior norm for a function is given by

${\displaystyle |u|_{k,\alpha$ ;\Omega }^{*}=|u|_{k;\Omega }^{*}+[u]_{k,\alpha ;\Omega }^{*}=\sum _{|\beta |\leq k}\sup _{x\in \Omega }|d_{x}^{|\beta |}D^{\beta }u(x)|+\sup _{\stackrel {x,y\in \Omega }{|\beta |=k}}d_{x,y}^{k+\alpha }{\frac {|D^{\beta }u(x)-D^{\beta }u(y)|}{|x-y|^{\alpha }}}}

It is occasionally necessary to add "extra" powers of the weight, denoted by

${\displaystyle |u|_{k,\alpha$ ;\Omega }^{(m)}=|u|_{k;\Omega }^{(m)}+[u]_{k,\alpha ;\Omega }^{(m)}=\sum _{|\beta |\leq k}\sup _{x\in \Omega }|d_{x}^{|\beta |+m}D^{\beta }u(x)|+\sup _{\stackrel {x,y\in \Omega }{|\beta |=k}}d_{x,y}^{m+k+\alpha }{\frac {|D^{\beta }u(x)-D^{\beta }u(y)|}{|x-y|^{\alpha }}}.}

Formulation

The formulations in this section are taken from the text of D. Gilbarg and Neil Trudinger (1983).

Sources

• Gilbarg, D.; Trudinger, Neil (1983), Elliptic Partial Differential Equations of Second Order, New York: Springer, ISBN 3-540-41160-7
• Schauder, Juliusz (1934), "Über lineare elliptische Differentialgleichungen zweiter Ordnung", Mathematische Zeitschrift (in German), Berlin, Germany: Springer-Verlag, vol. 38, no. 1, pp. 257–282, doi:10.1007/BF01170635, S2CID 120461752 MR1545448
• Schauder, Juliusz (1937), "Numerische Abschätzungen in elliptischen linearen Differentialgleichungen" (PDF), Studia Mathematica (in German), Lwów, Poland: Polska Akademia Nauk. Instytut Matematyczny, vol. 5, pp. 34–42

Further reading

• Courant, Richard; Hilbert, David (1989), Methods of Mathematical Physics, vol. 2 (1st English ed.), New York: Wiley-Interscience, ISBN 0-471-50439-4
• Han, Qing; Lin, Fanghua (1997), Elliptic Partial Differential Equations, New York: Courant Institute of Mathematical Sciences, ISBN 0-9658703-0-8, OCLC 38168365 MR1669352