Exercise 2.1.2 A Nonhyperbolic Cauchy Problem for the Wave Equation)
4.0 Results on elliptic boundary value problems
In this chapter we consider the elliptic second-order differential equation Au= f in, f = f0−
n j=1
∂jfj
(4.0.1)
with the Dirichlet boundary data
u=g0 on∂.
(4.0.2)
We assume that A=div(−a∇)+bã ∇ +cwith bounded and measurable coef- ficientsa (symmetric real-valued (n×n) matrix) and complex-valuedbandcin L∞(). Another assumption is thatAis an elliptic operator; i.e., there isε0>0 such thata(x)ξãξ ≥ε0|ξ|2for any vectorξ ∈Rnand anyx∈. Unless specified otherwise, we assume thatis a bounded domain inRnwith the boundary of class C2. However, most of the results are valid for Lipschitz boundaries. We recall that for anyu∈ H(1)() there are boundary valuesu,∇uon∂that are contained in H(1/2)(∂),H(−1/2)(∂). These boundary values can be understood via approxi- mations as follows. One can approximateuinH(1)() by functionsum∈C∞( ¯), and then boundary values are understood as the limits. From trace theorems it follows that um(1/2)(∂),∇um(−1/2)(∂) are bounded byCum(1)(). In particular,φm∂jum2(∂)≤Cφm(1)()um(1)(). Since approximations are convergent in H(1)(), they are fundamental in this space, and so limit of their boundary traces is the needed trace ofu. A (weak) solutionu∈ H(1)() is defined in the sense of the integral identity
(a∇uã ∇φ+bã ∇uφ+cuφ)−
∂∂ν(a)uφ
=
f0φ+
n j=1
fj∂jφ (4.0.3)
89
for any test functionφ∈ H(1)(), provided that the boundary Dirichlet condition (4.0.2) is satisfied and suppfj ⊂.
We will formulate basic results about solvability and regularity of the Dirichlet problem (4.0.1), (4.0.2).
Theorem 4.1. Let ∂be Lipschitz, g0∈ H(1/2)(∂), and f = f0+
j≤n∂jfj
with fj ∈L2(). Let us replace c by c+λ.
Then for all λ∈ E, where E is a set of points converging to ∞, there is a unique generalized solution u∈ H(1)()to the Dirichlet problem(4.0.1),(4.0.2).
Moreover,
u(1)()≤C
g0(1/2)(∂)+
0≤j≤n
fj2()+ u2()
, (4.0.4)
where C depends only on, λand the mentioned bounds on the coefficients of A.
If a solution is unique, then it does exist. Moreover, one can drop the last term in the previous bound. In particular, if c≥0,Ib=0, or if b=0,Rc≥0, then a solution exists and is unique.
If fj ∈L∞(0)for some0⊂, then u∈Cà(0)for someà∈(0,1). In addition, when¯0 ⊂and p>n, there is constant C depending on the same parameters as above such that
u(1)(01)+ u∞(01)≤C n
j=0
fjp(0)+ u2(0)
.
Here and below01 ⊂0is any domain with positive distance to∂0.
If a,b,c∈Cγ( ¯0), fj∈Cγ( ¯0), g0 ∈C1+γ(∂0),∂0∈C2+γ, then for any domain01 with positive distance to∂0∩there is a constant C depending only on01and on the norms of a,b,c in Cγ(0)such that
|u|1+γ(01)≤C
0≤j≤n
|fj|γ(0)+ |g0|1+γ(∂0∩∂)+ |u|0(0)
. (4.0.5)
If ∇a∈ L∞(0), f1= ã ã ã = fn =0 in 0, f ∈L∞(0), ∂0∈C2, g0∈ H(3/2)(∂0∩∂), then u∈ H(2)(01)for any01mentioned above.
If f1= ã ã ã = fn =0; a,∇a ∈C( ¯); b,c∈ L∞(); f ∈Lp(0), and g0∈ C2(∂0∩∂), then u∈Hp;k(01), and there is C depending on01,p, the norms of the coefficients in the above-mentioned spaces, and the ellipticity constant of A such that
u2,p(01)≤C(fp(0)+ |g0|2(∂∩∂0)+ u2(0)).
If f1= ã ã ã = fn =0; a,∇a,b,c, f ∈Cγ( ¯j∩0); ∂j∩¯0∈C2; ¯ =
∪¯j, wherej are some disjoint subdomains of, then u∈C1( ¯j∩¯01). If
4.0. Results on elliptic boundary value problems 91 in addition,∂j∩¯0∈C2+γ, then u∈C2+γ( ¯j∩¯01)and
|u|2+γ(01∩¯j)
≤C
j
|f|γ(0∩j)+ |g0|2+γ(∂∩∂0)+ u∞(0)
, where C depends on01,γ, the norms of coefficients in the above-mentioned H¨older spaces, and the ellipticity constant of A.
A proof of this result can be found in the book of Ladyzhenskaya and Ural’tseva [LU], pp. 149, 189, 202, 222, except of the bound (4.0.5) and the subsequent H2,p(01)-bound, which are obtained by Agmon, Douglis, and Nirenberg [ADN], and theL∞-bound in the case of divergent equations with measurable and bounded coefficients given by Kinderlehrer and Stampacchia [KinS]. Theorem 4.1 is in fact a formulation of several results of the available theory of elliptic boundary value problems for equations of second order that are sufficient for considering the inverse problems in this book. Of course, it is neither comprehensive nor most general. The domain0is needed to formulate local results when the data of the problem are regular only in a subdomain of. When=0=01, the results are more transparent. Observe that for Lipschitz a solution to the Dirichlet problem with smooth data can be not inH(2)().
Theorem 4.2(Comparison Principle). Assume thatIb=0, c≥0in.
If f1≤ f2inand g01≤g02, then for solutions u1,u2to the Dirichlet problem (4.0.1),(4.0.2)with the data f1,g01and f2,g02we have u1≤u2.
(Hopf Maximum Principle) If a solution u to equation (4.0.1) with f =0is in C( ¯)thenu∞()≤sup|g0|over∂. If f ≥0in, theninfu≥inf∂g0. (Giraud Maximum Principle) Assume in addition that∂∈C1+λand that a∈ C1+λ( ¯),b,c∈Cλ( ¯). Let l be any nontangential outward direction at x∈∂.
Assume that c≥0, f ≥0in.
If x is a maximum point of u∈C( ¯)inand u(x)>0, then there isε >0 such that
εt <u(x+tl)−u(x) when0<t < ε. When there is∂lu(x), then∂lu(x)>0.
A proof of the Hopf maximum principle fora ∈C1( ¯) and foru ∈C2( ¯) can be found in the book of Miranda [Mi], p. 6. In the slightly more general case we con- sider, it follows by using the approximation results in [LU], p. 158, which claims that if the coefficients of (4.0.1) are convergent almost everywhere, uniformly satisfying boundedness and ellipticity conditions, and if the source terms and the boundary conditions are convergent correspondingly inH(−1)() andH(1/2)(∂), the solutions to the Dirichlet problem (4.0.1), (4.0.2) are convergent in H(1)().
The inequality u≥0 for functions in H(1) is understood in the sense that such functions can beH(1)-approximated by nonnegative smooth functions. For maxi- mum principles for weak solutions we refer also to the book of Kinderlehrer and Stampacchia [KinS], p. 38.
It is easy to see that the Hopf maximum principle implies the comparison principle. To see this, observe that u=u2−u1 satisfies equation (4.0.1) with f = f2− f1≥0, so by the second part of the Hopf principle we conclude that u≥g02−g01≥0.
We are interested in findinga,b,cgiven the additional boundary data
∂ν(a)u=g1on, (4.0.6)
which is a part of∂. According to Theorem 4.0.1, ifa,b,c, f areC1-smooth near∂∈C2+λ and g0 ∈C1+λ(), then ∇u is continuous near ∂and has a continuous extension onto∪, so the Neumann data can be understood in the classical sense.