Differential equation, partial, free boundaries

The problem of finding the solution of a system of partial differential equations with suitable initial and boundary conditions in a domain whose boundary is completely, or partially, unknown and must be determined. Problems of this type arise in many problems of filtration, diffusion, thermal conduction, and other branches of the mechanics of continuous media. For instance, the Helmholtz–Kirchhoff problem on symmetrical current flow around an equilateral wedge with an apex angle of $ 2 \alpha \pi $ of an unbounded flux of an ideal liquid consists in finding the velocity components $ u = u ( x , y ) $, $ v = v ( x , y ) $, which are the solution of the Cauchy–Riemann system of equations:

$$ \frac{\partial u }{\partial x } + \frac{\partial v }{\partial y } = 0 ,\ \ \frac{\partial u }{\partial y } - \frac{\partial v }{\partial x } = 0 , $$

and which satisfy, on a certain boundary, the conditions (see Fig.):

$$ \left . v \right | _ {A O } = 0 ,\ \left . v \right | _ {O B } = \left . \mathop{\rm tan} \alpha \pi u \right | _ {O B } . $$

The unknown boundary $ BS $ is a stream line on which the supplementary condition

$$ u ^ {2} + v ^ {2} = v _ \infty ^ {2} ,\ \lim\limits _ {r \rightarrow \infty } u = v _ \infty ,\ r = \sqrt {x ^ {2} + y ^ {2} } , $$

has been imposed. The figure shows the upper half of the stream, $ y \geq 0 $; for $ y \leq 0 $ the stream is symmetric.

Figure: d031980a

In solving problems with a free boundary in the plane, extensive use is made of methods of the theory of functions of a complex variable.

See also Stefan problem.

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Free boundary problems for partial differential equations are frequently encountered in a variety of applications. Besides the classical free boundary problems in fluid dynamics (e.g., jets and cavities), in filtration theory (e.g. the displacement of immiscible fluids in porous media, also known as Muskat's or Verigin's problem), in change of phase (Stefan problem), in crystal growth, and in linear elasticity (e.g. the obstacle problem), many other interesting problems can be listed. For instance, free boundaries can arise in the motion of non-Newtonian fluids (A.E. Bingham), in non-linear diffusion with degeneracy (i.e. diffusion governed by the so-called porous medium equation), in reaction-diffusion systems, etc. A common feature of free boundary problems is the necessity of extra conditions on the free boundary. Such conditions involve the unknown function $ u $ in the partial differential equation and/or its derivatives up to an order which can even exceed the order of the differential equation. They may or may not involve the normal component of the velocity of the free boundary. Sometimes they have a non-local character, being expressed in terms of functionals acting on $ u $ and on the free boundary itself. From the mathematical point of view the occurrence of a free boundary can be frequently related to discontinuities or even singularities of some of the coefficients of the differential equation in correspondence of some given value $ u _ {0} $ of $ u $. In studying a free boundary problem of the latter type one can follow the classical approach, trying to determine the free boundary and a regular solution $ u $ of the partial differential equation, or one can consider a generalized (or weak) formulation in which the free boundary is defined implicitly as the set $ \{ u = u _ {0} \} $, requiring $ u $ to belong to some Sobolev space. However in such a case it is generally very hard to retrieve any information on the regularity of the free boundary.

See also Differential equation, partial, discontinuous coefficients; Differential equation, partial, with singular coefficients.

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