Jets and Cavities

1. Examples of jets and cavities. 266 2. The variational problem, 271 3. Regularity and nondegeneracy, 275 4. Regularity of the free boundary, 284 5. The bounded gradient lemma and the nonoscillation lemma, 285 6. Convergence of free boundaries, 288 7. Symmetric rearrangements, 293 8. Axially symmetric jet flows, 297 9. The free boundary is a curve x = k(y). 309

10. Monotonicity and uniqueness, 318 11. The smooth-fit theorems, 324 12. Existence and uniqueness for axially symmetric jets, 335 13. Convexity of the free boundary. 339 14. The plane symmetric jet flow, 344 15. Asymmetric jet flows, 345 16. The free boundary for the asymmetric case, 353 17. Monotonicity, continuity, and existence for the asymmetric

jet problem, 358 18. Jets with gravity, 366 19. The continuous fit for the gravity case, 379 20. Axially symmetric cavities, 391 21. Axially symmetric infinite cavities, 400 22. Bibliographical remarks, 415

4. VarIational Problems with Potentials 417

I. Self-gravitating axisymmetric rotating fluids, 418 2. Estimates of gravitational potentials, 426 3. Existence of solutions, 433 4. Rapidly rotating 443 5. The rings of rotating fluids, 456 6. Vortex rings, 470 7. Energy identities and potential estimates, 478 8. Existence of vortex rings, 487 9. A capacty estimate, 501

10. Asymptotic estimates for 507 11. The plasma qi solutioee, 520 12. The free boundary for the plasma problem, 529 13. Asymptotic estimates for the problem, 534

’14. A variational approach to convex plasmas, 549 15. The Thomas-Fermi model,. 563 16. Existence of solution for the 56R 17. Regularity of the free boundary for the Thomas-Fermi 576 18. Bibliographical remarks, 586


5. Some Free-boundary Problems Not in Variational Form 589

I. The porous-medium equation: existence and uniqueness, 589 2. Estimates on the expansion of gas, 604 3. Holder continuity of the solution, 617 4. Growth and HOlder continuity of the free boundary, 627 5. The differential equation on the free boundary, 636 6. The general two-dimensional filtration problem: existence, 650 7. Regularity of the free boundary, 657 8. Uniqueness for the filtration problem, 668 9. The filtration problem in n dimensions, 674

10. The two-phase Stefan problem, 684 11. Bibliographical remarks, 693

References 695

Index 709


The Dirichiet problem for the Laplace operator seeks a solution ii of u 4) on the boundary Suppose that only a

portion S of is given whereas the remaining portion r not a priori prescribed, and an additional condition is imposed on the unknown part of the boundary. such as V(u — 4)) = 0 on I’ (here 4) is a given function in the entire space). Thus we seek to determine u and r satisfying


where D is bounded bX the given S and the unknown r. This problem is an example of a free-boundary problem.

For a two-dimensional ideal fluid, the density function u satisfies, on the interface F between the fluid and the air, the free-boundary conditions

u = C1, I Vu C2 (C1, C2 constants),

where either C1 or C2 is frequently unknown; F is also not prescribed. Another example of a free-boundary problem occurs when ice and water

share a common interface. Here the free-boundary conditions are

0, — =

where is the tàmperature in the water, °2 is the temperature in the ice, a and k are positive constants, •(x, 1) = 0 describes the eqtzatibñ of thern free boundary, ana 0, satisfies the paiabolic equation

There are also free-boundary problems in which the freeboundary not appear explicitly the outset of the problem. example, the ol axisymmetric self-gravitating rotatiftg fluid the boundary separating the fluid from the vacuum as the set



to a nonlinear equation

(‘(U (c>O,8>O)

u is “potential’ function depending on the fluid’s density p and (u = 0) = p For in a medium, the free boundary is the boundary u where ii the density satisfying the nonlinear degenerate parabolic


u,=,&um (m>l).

the initial step in studying a free-boundary problem is to refor- mulate it in such a that the free boundary disappears. (There are, however, notable exceptions, especially in the case of one space dimension.) Such reformulations can often he achieved by resorting to variational principles. One of the most celebrated examples is the following: If u minimizes

J(t) =f[i Vv12 + 2fv1

subject () on afl. v in then u satisfies at least)

u is a solution of the free-boundary problem of the’ type mentioned earlier.

Since the problem of minimizing J( v) has a solution (obtained as a limit of a minimizing sequence). we conclude that there exists a solution u of the free-boundary problem in its variational formulation.

The next two steps after establishing the existence of a solution for the reformulated free-boundary problem are to obtain the best regularity results and then proceed to analyze the free bottndary. The latter step often requires much deeper methods.

Thus in the preceding problem (*), the optimal regularity is that u has Lipschitz continuous first derivatives. There are also known sufficient condi- (ions that ensure that the free boundary is smooth, but in general (without such conditions) the free boundary may be quite singular.

in Chapters 1 and 2 we develop the theory of a large class of free-boundary problems called variational inequalities. Chapter 1 deals with the variational


approach, existence, uniqueness. and regularity of the minimizer. Chapter 2 is concerned with the study of the free boundary itself.

In Chapter 3 we study a class of variational problems designed for solving problems of jets and cavities of ideal fluids. Whereas in Chapter 1 a typical functional is

f[$vvI2 + 2fvJ (r>O)

in Chapter 3 the corresponding functional is

where ‘A is the characteristic function of A. The variational functional in Chapter 4 is of the type


where p is a density function subject to some constraints. Here the free boundary is 8(p > 0).

In Chapter 5 we study several free-boundary problems that are not for- rnulted as variational problems; we deal mainly with gas ;n a porous medium and with the filtration of fluid in a porous dam.

Chapters 3, 4, and 5 are basically independent of each other as far as cross references are concerned. However, they do share some common methodL techniques, and ideas. The material of Chapters 1 and 2 appears in later chapters either directly or indirectly.

There is a large body of literature on time-dependent fke-boundary prob- lems in one space dimension. Here the methods are often highly specialized. With a few exceptions, we shall not deal with such problems in the present book.



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