CHAPTER 4. Elliptic PDEs

Transcription

1 CHAPTER 4 Elliptic PDEs One of te main advantages of extending te class of solutions of a PDE from classical solutions wit continuous derivatives to weak solutions wit weak derivatives is tat it is easier to prove te existence of weak solutions. Having establised te existence of weak solutions, one may ten study teir properties, suc as uniqueness and regularity, and peraps prove under appropriate assumptions tat te weak solutions are, in fact, classical solutions. Tere is often considerable freedom in ow one defines a weak solution of a PDE; for example, te function space to wic a solution is required to belong is not given a priori by te PDE itself. Typically, we look for a weak formulation tat reduces to te classical formulation under appropriate smootness assumptions and wic is amenable to a matematical analysis; te notion of solution and te spaces to wic solutions belong are dictated by te available estimates and analysis Weak formulation of te Diriclet problem Let us consider te Diriclet problem for te Laplacian wit omogeneous boundary conditions on a bounded domain in R n, (4.1) (4.2) u = f in, u = on. First, suppose tat te boundary of is smoot and u, f : R are smoot functions. Multiplying (4.1) by a test function φ, integrating te result over, and using te divergence teorem, we get (4.3) Du Dφ dx = fφ dx for all φ Cc (). Te boundary terms vanis because φ = on te boundary. Conversely, if f and are smoot, ten any smoot function u tat satisfies (4.3) is a solution of (4.1). Next, we formulate weaker assumptions under wic (4.3) makes sense. We use te flexibility of coice to define weak solutions wit L 2 -derivatives tat belong to a Hilbert space; tis is elpful because Hilbert spaces are easier to work wit tan Banac spaces. 1 It also leads to a variational form of te equation tat is symmetric in te solution u and te test function φ. By te Caucy-Scwartz inequality, te integral on te left-and side of (4.3) is finite if Du belongs to L 2 (), so we suppose tat u H 1 (). We impose te boundary condition (4.2) in a weak sense by requiring tat u H 1 (). Te left and side of (4.3) ten extends by continuity to φ H 1 () = Cc (). 1 We would need to use Banac spaces to study te solutions of Laplace s equation wose derivatives lie in L p for p 2, and we may be forced to use Banac spaces for some PDEs, especially if tey are nonlinear. 87

2 88 4. ELLIPTIC PDES Te rigt and side of (4.3) is well-defined for all φ H 1 () if f L 2 (), but tis is not te most general f for wic it makes sense; we can define te rigt-and for any f in te dual space of H 1 (). Definition 4.1. Te space of bounded linear maps f : H 1 () R is denoted by H 1 () = H 1 (), and te action of f H 1 () on φ H 1 () by f, φ. Te norm of f H 1 () is given by { } f, φ f H 1 = sup : φ H 1, φ. φ H 1 A function f L 2 () defines a linear functional F f H 1 () by F f, v = fv dx = (f, v) L 2 for all v H 1 (). Here, (, ) L 2 denotes te standard inner product on L 2 (). Te functional F f is bounded on H 1 () wit F f H 1 f L 2 since, by te Caucy-Scwartz inequality, F f, v f L 2 v L 2 f L 2 v H 1. We identify F f wit f, and write bot simply as f. Suc linear functionals are, owever, not te only elements of H 1 (). As we will sow below, H 1 () may be identified wit te space of distributions on tat are sums of first-order distributional derivatives of functions in L 2 (). Tus, after identifying functions wit regular distributions, we ave te following triple of Hilbert spaces H 1 () L 2 () H 1 (), H 1 () = H 1 (). Moreover, if f L 2 () H 1 () and u H 1 (), ten f, u = (f, u) L 2, so te duality pairing coincides wit te L 2 -inner product wen bot are defined. Tis discussion motivates te following definition. Definition 4.2. Let be an open set in R n and f H 1 (). A function u : R is a weak solution of (4.1) (4.2) if: (a) u H 1 (); (b) (4.4) Du Dφ dx = f, φ for all φ H 1 (). Here, strictly speaking, function means an equivalence class of functions wit respect to pointwise a.e. equality. We ave assumed omogeneous boundary conditions to simplify te discussion. If is smoot and g : R is a function on te boundary tat is in te range of te trace map T : H 1 () L 2 ( ), say g = T w, ten we obtain a weak formulation of te nonomogeneous Diricet problem u = f in, u = g on, by replacing (a) in Definition 4.2 wit te condition tat u w H 1 (). Te definition is oterwise te same. Te range of te trace map on H 1 () for a smoot domain is te fractional-order Sobolev space H 1/2 ( ); tus if te boundary data g is so roug tat g / H 1/2 ( ), ten tere is no solution u H 1 () of te nonomogeneous BVP.

3 4.2. VARIATIONAL FORMULATION Variational formulation Definition 4.2 of a weak solution in is closely connected wit te variational formulation of te Diriclet problem for Poisson s equation. To explain tis connection, we first summarize some definitions of te differentiability of functionals (scalar-valued functions) acting on a Banac space. Definition 4.3. A functional J : X R on a Banac space X is differentiable at x X if tere is a bounded linear functional A : X R suc tat J(x + ) J(x) A lim =. X If A exists, ten it is unique, and it is called te derivative, or differential, of J at x, denoted DJ(x) = A. Tis definition expresses te basic idea of a differentiable function as one wic can be approximated locally by a linear map. If J is differentiable at every point of X, ten DJ : X X maps x X to te linear functional DJ(x) X tat approximates J near x. A weaker notion of differentiability (even for functions J : R 2 R see Example 4.4) is te existence of directional derivatives [ ] J(x + ɛ) J(x) δj(x; ) = lim = d ɛ ɛ dɛ ɛ= J(x + ɛ). If te directional derivative at x exists for every X and is a bounded linear functional on, ten δj(x; ) = δj(x) were δj(x) X. We call δj(x) te Gâteaux derivative of J at x. Te derivative DJ is ten called te Frécet derivative to distinguis it from te directional or Gâteaux derivative. If J is differentiable at x, ten it is Gâteaux-differentiable at x and DJ(x) = δj(x), but te converse is not true. Example 4.4. Define f : R 2 R by f(, ) = and ( ) xy 2 2 f(x, y) = x 2 + y 4 if (x, y) (, ). Ten f is Gâteaux-differentiable at, wit δf() =, but f is not Frécetdifferentiable at. If J : X R attains a local minimum at x X and J is differentiable at x, ten for every X te function J x; : R R defined by J x; (t) = J(x + t) is differentiable at t = and attains a minimum at t =. It follows tat dj x; () = δj(x; ) = for every X. dt Hence DJ(x) =. Tus, just as in multivariable calculus, an extreme point of a differentiable functional is a critical point were te derivative is zero. Given f H 1 (), define a quadratic functional J : H 1 () R by (4.5) J(u) = 1 Du 2 dx f, u. 2 Clearly, J is well-defined.

4 9 4. ELLIPTIC PDES Proposition 4.5. Te functional J : H 1 () R in (4.5) is differentiable. Its derivative DJ(u) : H 1 () R at u H 1 () is given by DJ(u) = Du D dx f, for H 1 (). Proof. Given u H 1 (), define te linear map A : H 1 () R by A = Du D dx f,. Ten A is bounded, wit A Du L 2 + f H 1, since A Du L 2 D L 2 + f H 1 H 1 ( Du L 2 + f H 1) H 1. For H 1 (), we ave It follows tat and terefore J(u + ) J(u) A = 1 2 D 2 dx. J(u + ) J(u) A H, 1 J(u + ) J(u) A lim =, H 1 wic proves tat J is differentiable on H 1 () wit DJ(u) = A. Note tat DJ(u) = if and only if u is a weak solution of Poisson s equation in te sense of Definition 4.2. Tus, we ave te following result. Corollary 4.6. If J : H 1 () R defined in (4.5) attains a minimum at u H 1 (), ten u is a weak solution of u = f in te sense of Definition 4.2. In te direct metod of te calculus of variations, we prove te existence of a minimizer of J by sowing tat a minimizing sequence {u n } converges in a suitable sense to a minimizer u. Tis minimizer is ten a weak solution of (4.1) (4.2). We will not follow tis metod ere, and instead establis te existence of a weak solution by use of te Riesz representation teorem. Te Riesz representation teorem is, owever, typically proved by a similar argument to te one used in te direct metod of te calculus of variations, so in essence te proofs are equivalent Te space H 1 () Te negative order Sobolev space H 1 () can be described as a space of distributions on. Teorem 4.7. Te space H 1 () consists of all distributions f D () of te form (4.6) f = f + i f i were f, f i L 2 (). Tese distributions extend uniquely by continuity from D() to bounded linear functionals on H 1 (). Moreover, ( n (4.7) f H 1 () = inf 1/2 fi dx) 2 : suc tat f, f i satisfy (4.6). i=

5 4.3. THE SPACE H 1 () 91 Proof. First suppose tat f H 1 (). By te Riesz representation teorem tere is a function g H 1 () suc tat (4.8) f, φ = (g, φ) H 1 for all φ H 1 (). Here, (, ) H 1 denotes te standard inner product on H 1 (), (u, v) H 1 = (uv + Du Dv) dx. Identifying a function g L 2 () wit its corresponding regular distribution, restricting f to φ D() H 1 (), and using te definition of te distributional derivative, we ave f, φ = gφ dx + i g i φ dx = g, φ + = g i g, i φ i g i, φ for all φ D(), were g i = i g L 2 (). Tus te restriction of every f H 1 () from H 1 () to D() is a distribution f = g i g i of te form (4.6). Also note tat taking φ = g in (4.8), we get f, g = g 2, H 1 wic implies tat ( 1/2 f H 1 g H 1 = g 2 dx + gi dx) 2, wic proves inequality in one direction of (4.7). Conversely, suppose tat f D () is a distribution of te form (4.6). Ten, using te definition of te distributional derivative, we ave for any φ D() tat f, φ = f, φ + i f i, φ = f, φ f i, i φ. Use of te Caucy-Scwartz inequality gives ( ) 1/2 f, φ f, φ 2 + f i, i φ 2. Moreover, since te f i are regular distributions belonging to L 2 () ( ) 1/2 ( 1/2 f i, i φ = f i i φ dx fi 2 dx i φ dx) 2, so [ ( ) ( ) ( ) ( ) ] 1/2 f, φ f 2 dx φ 2 dx + fi 2 dx i φ 2 dx,

6 92 4. ELLIPTIC PDES and ( ) 1/2 ( f, φ f 2 dx + fi 2 dx φ 2 + ( n 1/2 fi dx) 2 φ H 1 i= ) 1/2 i φ 2 dx Tus te distribution f : D() R is bounded wit respect to te H 1 ()-norm on te dense subset D(). It terefore extends in a unique way to a bounded linear functional on H 1 (), wic we still denote by f. Moreover, ( n 1/2 f H 1 fi dx) 2, i= wic proves inequality in te oter direction of (4.7). Te dual space of H 1 () cannot be identified wit a space of distributions on because D() is not a dense subspace. Any linear functional f H 1 () defines a distribution by restriction to D(), but te same distribution arises from different linear functionals. Conversely, any distribution T D () tat is bounded wit respect to te H 1 -norm extends uniquely to a bounded linear functional on H 1, but te extension of te functional to te ortogonal complement (H 1 ) in H 1 is arbitrary (subject to maintaining its boundedness). Rougly speaking, distributions are defined on functions wose boundary values or trace is zero, but general linear functionals on H 1 depend on te trace of te function on te boundary. Example 4.8. Te one-dimensional Sobolev space H 1 (, 1) is imbedded in te space C([, 1]) of continuous functions, since p > n for p = 2 and n = 1. In fact, according to te Sobolev imbedding teorem H 1 (, 1) C,1/2 ([, 1]), as can be seen directly from te Caucy-Scwartz inequality: f(x) f(y) x y ( x y f (t) dt ) 1/2 ( x ) 1/2 1 dt f (t) 2 dt y ( 1 1/2 f (t) dt) 2 x y 1/2. As usual, we identify an element of H 1 (, 1) wit its continuous representative in C([, 1]). By te trace teorem, Te ortogonal complement is H 1 (, 1) = { u H 1 (, 1) : u() = u(1) }. H 1 (, 1) = { u H 1 (, 1) : suc tat (u, v) H 1 = for every v H 1 (, 1) }. Tis condition implies tat u H 1 (, 1) if and only if 1 (uv + u v ) dx = for all v H 1 (, 1),

7 4.4. THE POINCARÉ INEQUALITY FOR H1 () 93 wic means tat u is a weak solution of te ODE It follows tat u(x) = c 1 e x + c 2 e x, so u + u =. H 1 (, 1) = H 1 (, 1) E were E is te two dimensional subspace of H 1 (, 1) spanned by te ortogonal vectors {e x, e x }. Tus, H 1 (, 1) = H 1 (, 1) E. If f H 1 (, 1) and u = u + c 1 e x + c 2 e x were u H 1 (, 1), ten f, u = f, u + a 1 c 1 + a 2 c 2 were f H 1 (, 1) is te restriction of f to H 1 (, 1) and a 1 = f, e x, a 2 = f, e x. Te constants a 1, a 2 determine ow te functional f H 1 (, 1) acts on te boundary values u(), u(1) of a function u H 1 (, 1) Te Poincaré inequality for H 1 () We cannot, in general, estimate a norm of a function in terms of a norm of its derivative since constant functions ave zero derivative. Suc estimates are possible if we add an additional condition tat eliminates non-zero constant functions. For example, we can require tat te function vanises on te boundary of a domain, or tat it as zero mean. We typically also need some sort of boundedness condition on te domain of te function, since even if a function vanises at some point we cannot expect to estimate te size of a function over arbitrarily large distances by te size of its derivative. Te resulting inequalities are called Poincaré inequalities. Te inequality we prove ere is a basic example of a Poincaré inequality. We say tat an open set in R n is bounded in some direction if tere is a unit vector e R n and constants a, b suc tat a < x e < b for all x. Teorem 4.9. Suppose tat is an open set in R n tat is bounded is some direction. Ten tere is a constant C suc tat (4.9) u 2 dx C Du 2 dx for all u H 1 (). Proof. Since Cc () is dense in H 1 (), it is sufficient to prove te inequality for u Cc (). Te inequality is invariant under rotations and translations, so we can assume witout loss of generality tat te domain is bounded in te x n - direction and lies between < x n < a. Writing x = (x, x n ) were x = (x 1,...,, x n 1 ), we ave xn a u(x, x n ) = n u(x, t) dt n u(x, t) dt. Te Caucy-Scwartz inequality implies tat a n u(x, t) dt = a ( a 1/2 1 n u(x, t) dt a 1/2 n u(x, t) dt) 2.

8 94 4. ELLIPTIC PDES Hence, u(x, x n ) 2 a a n u(x, t) 2 dt. Integrating tis inequality wit respect to x n, we get a a u(x, x n ) 2 dx n a 2 n u(x, t) 2 dt. A furter integration wit respect to x gives u(x) 2 dx a 2 n u(x) 2 dx. Since n u Du, te result follows wit C = a 2. Tis inequality implies tat we may use as an equivalent inner-product on H 1 an expression tat involves only te derivatives of te functions and not te functions temselves. Corollary 4.1. If is an open set tat is bounded in some direction, ten H 1 () equipped wit te inner product (4.1) (u, v) = Du Dv dx is a Hilbert space, and te corresponding norm is equivalent to te standard norm on H 1 (). Proof. We denote te norm associated wit te inner-product (4.1) by ( 1/2 u = Du dx) 2, and te standard norm and inner product by ( [ u 1 = u 2 + Du 2] 1/2 dx), (4.11) (u, v) 1 = (uv + Du Dv) dx. Ten, using te Poincaré inequality (4.9), we ave u u 1 (C + 1) 1/2 u. Tus, te two norms are equivalent; in particular, (H 1, (, ) ) is complete since (H 1, (, ) 1 ) is complete, so it is a Hilbert space wit respect to te inner product (4.1) Existence of weak solutions of te Diriclet problem Wit tese preparations, te existence of weak solutions is an immediate consequence of te Riesz representation teorem. Teorem Suppose tat is an open set in R n tat is bounded in some direction and f H 1 (). Ten tere is a unique weak solution u H 1 () of u = f in te sense of Definition 4.2.

9 4.5. EXISTENCE OF WEAK SOLUTIONS OF THE DIRICHLET PROBLEM 95 Proof. We equip H 1 () wit te inner product (4.1). Ten, since is bounded in some direction, te resulting norm is equivalent to te standard norm, and f is a bounded linear functional on ( ) H 1 (), (, ). By te Riesz representation teorem, tere exists a unique u H 1 () suc tat (u, φ) = f, φ for all φ H 1 (), wic is equivalent to te condition tat u is a weak solution. Te same approac works for oter symmetric linear elliptic PDEs. Let us give some examples. Example Consider te Diriclet problem u + u = f in, u = Ten u H 1 () is a weak solution if (Du Dφ + uφ) dx = f, φ Tis is equivalent to te condition tat (u, φ) 1 = f, φ on. for all φ H 1 (). for all φ H 1 (). were (, ) 1 is te standard inner product on H 1 () given in (4.11). Tus, te Riesz representation teorem implies te existence of a unique weak solution. Note tat in tis example and te next, we do not use te Poincaré inequality, so te result applies to arbitrary open sets, including = R n. In tat case, H 1 (R n ) = H 1 (R n ), and we get a unique solution u H 1 (R n ) of u + u = f for every f H 1 (R n ). Moreover, using te standard norms, we ave u H 1 = f H 1. Tus te operator + I is an isometry of H 1 (R n ) onto H 1 (R n ). Example As a sligt generalization of te previous example, suppose tat µ >. A function u H 1 () is a weak solution of (4.12) u + µu = f in, u = on. if (u, φ) µ = f, φ for all φ H 1 () were (u, v) µ = (µuv + Du Dv) dx Te norm µ associated wit tis inner product is equivalent to te standard one, since 1 C u 2 µ u 2 1 C u 2 µ were C = max{µ, 1/µ}. We terefore again get te existence of a unique weak solution from te Riesz representation teorem. Example Consider te last example for µ <. If we ave a Poincaré inequality u L 2 C Du 2 L for, wic is te case if is bounded in some direction, ten ( (u, u) µ = µu 2 + Du Dv ) dx (1 C µ ) Du 2 dx.

10 96 4. ELLIPTIC PDES Tus u µ defines a norm on H 1 () tat is equivalent to te standard norm if 1/C < µ <, and we get a unique weak solution in tis case also. For bounded domains, te Diriclet Laplacian as an infinite sequence of real eigenvalues {λ n : n N} suc tat tere exists a nonzero solution u H 1 () of u = λ n u. Te best constant in te Poincaré inequality can be sown to be te minimum eigenvalue λ 1, and tis metod does not work if µ λ 1. For µ = λ n, a weak solution of (4.12) does not exist for every f H 1 (), and if one does exist it is not unique since we can add to it an arbitrary eigenfunction. Tus, not only does te metod fail, but te conclusion of Teorem 4.11 may be false. Example Consider te second order PDE i (a ij j u) = f in, (4.13) u = on were te coefficient functions a ij : R are symmetric (a ij = a ji ), bounded, and satisfy te uniform ellipticity condition tat for some θ > a ij (x)ξ i ξ j θ ξ 2 for all x and all ξ R n. Also, assume tat is bounded in some direction. Ten a weak formulation of (4.13) is tat u H 1 () and a(u, φ) = f, φ for all φ H 1 (), were te symmetric bilinear form a : H 1 () H 1 () R is defined by a(u, v) = a ij i u j v dx. Te boundedness of a ij, te uniform ellipticity condition, and te Poincaré inequality imply tat a defines an inner product on H 1 wic is equivalent to te standard one. An application of te Riesz representation teorem for te bounded linear functionals f on te Hilbert space (H 1, a) ten implies te existence of a unique weak solution. We discuss a generalization of tis example in greater detail in te next section General linear, second order elliptic PDEs Consider PDEs of te form Lu = f were L is a linear differential operator of te form (4.14) Lu = i (a ij j u) + i (b i u) + cu, acting on functions u : R were is an open set in R n. A pysical interpretation of suc PDEs is described briefly in Section 4.A. We assume tat te given coefficients functions a ij, b i, c : R satisfy (4.15) a ij, b i, c L (), a ij = a ji.

11 4.6. GENERAL LINEAR, SECOND ORDER ELLIPTIC PDES 97 Te operator L is elliptic if te matrix (a ij ) is positive definite. We will assume te stronger condition of uniformly ellipticity given in te next definition. Definition Te operator L in (4.14) is uniformly elliptic on if tere exists a constant θ > suc tat (4.16) a ij (x)ξ i ξ j θ ξ 2 for x almost everywere in and every ξ R n. Tis uniform ellipticity condition allows us to estimate te integral of Du 2 in terms of te integral of a ij i u j u. Example Te Laplacian operator L = is uniformly elliptic on any open set, wit θ = 1. Example Te Tricomi operator L = y 2 x + 2 y is elliptic in y > and yperbolic in y <. For any < ɛ < 1, L is uniformly elliptic in te strip {(x, y) : ɛ < y < 1}, wit θ = ɛ, but it is not uniformly elliptic in {(x, y) : < y < 1}. For µ R, we consider te Diriclet problem for L + µi, Lu + µu = f in, (4.17) u = on. We motivate te definition of a weak solution of (4.17) in a similar way to te motivation for te Laplacian: multiply te PDE by a test function φ Cc (), integrate over, and use integration by parts, assuming tat all functions and te domain are smoot. Note tat i (b i u)φ dx = b i u i φ dx. Tis leads to te condition tat u H 1 () is a weak solution of (4.17) wit L given by (4.14) if a ij i u j φ b i u i φ + cuφ dx + µ uφ dx = f, φ for all φ H 1 (). To write tis condition more concisely, we define a bilinear form by (4.18) a(u, v) = a : H 1 () H 1 () R a ij i u j v b i u i v + cuv dx. Tis form is well-defined and bounded on H 1 (), as we ceck explicitly below. We denote te L 2 -inner product by (u, v) L 2 = uv dx. i

12 98 4. ELLIPTIC PDES Definition Suppose tat is an open set in R n, f H 1 (), and L is a differential operator (4.14) wose coefficients satisfy (4.15). Ten u : R is a weak solution of (4.17) if: (a) u H 1 (); (b) a(u, φ) + µ(u, φ) L 2 = f, φ for all φ H 1 (). Te form a in (4.18) is not symmetric unless b i =. We ave a(v, u) = a (u, v) were (4.19) a (u, v) = a ij i u j v + b i ( i u)v + cuv dx i is te bilinear form associated wit te formal adjoint L of L, (4.2) L u = i (a ij j u) b i i u + cu. Te proof of te existence of a weak solution of (4.17) is similar to te proof for te Diriclet Laplacian, wit one exception. If L is not symmetric, we cannot use a to define an equivalent inner product on H 1 () and appeal to te Riesz representation teorem. Instead we use a result due to Lax and Milgram wic applies to non-symmetric bilinear forms Te Lax-Milgram teorem and general elliptic PDEs We begin by stating te Lax-Milgram teorem for a bilinear form on a Hilbert space. Afterwards, we verify its ypoteses for te bilinear form associated wit a general second-order uniformly elliptic PDE and use it to prove te existence of weak solutions. Teorem 4.2. Let H be a Hilbert space wit inner-product (, ) : H H R, and let a : H H R be a bilinear form on H. Assume tat tere exist constants C 1, C 2 > suc tat C 1 u 2 a(u, u), a(u, v) C 2 u v for all u, v H. Ten for every bounded linear functional f : H R, tere exists a unique u H suc tat f, v = a(u, v) for all v H. For te proof, see [5]. Te verification of te ypoteses for (4.18) depends on te following energy estimates. 2 Te story beind tis result te story migt be completely true or completely false is tat Lax and Milgram attended a seminar were te speaker proved existence for a symmetric PDE by use of te Riesz representation teorem, and one of tem asked te oter if symmetry was required; in alf an our, tey convinced temselves tat is wasn t, giving birt to te Lax- Milgram lemma.

13 4.7. THE LAX-MILGRAM THEOREM AND GENERAL ELLIPTIC PDES 99 Teorem Let a be te bilinear form on H 1 () defined in (4.18), were te coefficients satisfy (4.15) and te uniform ellipticity condition (4.16) wit constant θ. Ten tere exist constants C 1, C 2 > and γ R suc tat for all u, v H 1 () (4.21) C 1 u 2 H a(u, u) + γ u 2 1 L 2 (4.22) a(u, v) C 2 u H1 v H 1, If b =, we may take γ = θ c were c = inf c, and if b, we may take γ = 1 b i 2 L 2θ + θ 2 c. Proof. First, we ave for any u, v H 1 () tat a(u, v) a ij i u j v dx + b i u i v dx + a ij L i u L 2 j v L 2 + C b i L u L 2 i v L 2 + c L u L 2 v L 2 a ij L + cuv dx. b i L + c L u H 1 v H 1, wic sows (4.22). Second, using te uniform ellipticity condition (4.16), we ave θ Du 2 L = θ Du 2 dx 2 a(u, u) + a(u, u) + a(u, u) + a ij i u j u dx b i u i u dx cu 2 dx b i u i u dx c u 2 dx b i L u L 2 i u L 2 c u L 2 a(u, u) + β u L 2 Du L 2 c u L 2, were c(x) c a.e. in, and ( n ) 1/2 β = b i 2 L. If β =, we get (4.21) wit γ = θ c, C 1 = θ.

14 1 4. ELLIPTIC PDES If β >, by Caucy s inequality wit ɛ, we ave for any ɛ > tat Hence, coosing ɛ = θ/2β, we get and (4.21) follows wit u L 2 Du L 2 ɛ Du 2 L ɛ u 2 L 2. θ 2 Du 2 L2 a(u, u) + ( ) β 2 2θ c u L 2, γ = β2 2θ + θ 2 c, C 1 = θ 2. Equation (4.21) is called Gårding s inequality; tis estimate of te H 1 -norm of u in terms of a(u, u), using te uniform ellipticity of L, is te crucial energy estimate. Equation (4.22) states tat te bilinear form a is bounded on H 1. Te expression for γ in tis Teorem is not necessarily sarp. For example, as in te case of te Laplacian, te use of Poincaré s inequality gives smaller values of γ for bounded domains. Teorem Suppose tat is an open set in R n, and f H 1 (). Let L be a differential operator (4.14) wit coefficients tat satisfy (4.15), and let γ R be a constant for wic Teorem 4.21 olds. Ten for every µ γ tere is a unique weak solution of te Diriclet problem in te sense of Definition Lu + µf =, u H 1 () Proof. For µ R, define a µ : H 1 () H 1 () R by (4.23) a µ (u, v) = a(u, v) + µ(u, v) L 2 were a is defined in (4.18). Ten u H 1 () is a weak solution of Lu + µu = f if and only if From (4.22), a µ (u, φ) = f, φ for all φ H 1 (). a µ (u, v) C 2 u H 1 v H 1 + µ u L 2 v L 2 (C 2 + µ ) u H 1 v H 1 so a µ is bounded on H 1 (). From (4.21), C 1 u 2 H 1 a(u, u) + γ u 2 L 2 a µ(u, u) wenever µ γ. Tus, by te Lax-Milgram teorem, for every f H 1 () tere is a unique u H 1 () suc tat f, φ = a µ (u, φ) for all v H 1 (), wic proves te result. Altoug L is not of exactly te same form as L, since it first derivative term is not in divergence form, te same proof of te existence of weak solutions for L applies to L wit a in (4.18) replaced by a in (4.19).

15 4.8. COMPACTNESS OF THE RESOLVENT Compactness of te resolvent An elliptic operator L + µi of te type studied above is a bounded, invertible linear map from H 1 () onto H 1 () for sufficiently large µ R, so we may define an inverse operator K = (L + µi) 1. If is a bounded open set, ten te Sobolev imbedding teorem implies tat H 1 () is compactly imbedded in L 2 (), and terefore K is a compact operator on L 2 (). Te operator (L λi) 1 is called te resolvent of L, so tis property is sometimes expressed by saying tat L as compact resolvent. As discussed in Example 4.14, L + µi may fail to be invertible at smaller values of µ, suc tat λ = µ belongs to te spectrum σ(l) of L, and te resolvent is not defined as a bounded operator on L 2 () for λ σ(l). Te compactness of te resolvent of elliptic operators on bounded open sets as several important consequences for te solvability of te elliptic PDE and te spectrum of te elliptic operator. Before describing some of tese, we discuss te resolvent in more detail. From Teorem 4.22, for µ γ we can define K : L 2 () L 2 (), K = (L + µi) 1 L 2 (). We define te inverse K on L 2 (), rater tan H 1 (), in wic case its range is a subspace of H 1 (). If te domain is sufficiently smoot for elliptic regularity teory to apply, ten u H 2 () if f L 2 (), and te range of K is H 2 () H 1 (); for non-smoot domains, te range of K is more difficult to describe. If we consider L as an operator acting in L 2 (), ten te domain of L is D = ran K, and L : D L 2 () L 2 () is an unbounded linear operator wit dense domain D. Te operator L is closed, meaning tat if {u n } is a sequence of functions in D suc tat u n u and Lu n f in L 2 (), ten u D and Lu = f. By using te resolvent, we can replace an analysis of te unbounded operator L by an analysis of te bounded operator K. If f L 2 (), ten f, v = (f, v) L 2. It follows from te definition of weak solution of Lu + µu = f tat (4.24) Kf = u if and only if a µ (u, v) = (f, v) L 2 for all v H 1 () were a µ is defined in (4.23). We also define te operator meaning tat K : L 2 () L 2 (), K = (L + µi) 1 L2 (), (4.25) K f = u if and only if a µ(u, v) = (f, v) L 2 for all v H 1 () were a µ(u, v) = a (u, v) + µ (u, v) L 2 and a is given in (4.19). Teorem If K B ( L 2 () ) is defined by (4.24), ten te adjoint of K is K defined by (4.25). If is a bounded open set, ten K is a compact operator. Proof. If f, g L 2 () and Kf = u, K g = v, ten using (4.24) and (4.25), we get (f, K g) L 2 = (f, v) L 2 = a µ (u, v) = a µ(v, u) = (g, u) L 2 = (u, g) L 2 = (Kf, g) L 2. Hence, K is te adjoint of K.

16 12 4. ELLIPTIC PDES If Kf = u, ten (4.21) wit µ γ and (4.24) imply tat C 1 u 2 H a 1 µ (u, u) = (f, u) L 2 f L 2 u L 2 f L 2 u H 1. Hence Kf H 1 C f L 2 were C = 1/C 1. It follows tat K is compact if is bounded, since it maps bounded sets in L 2 () into bounded sets in H 1 (), wic are precompact in L 2 () by te Sobolev imbedding teorem. Consider te Diriclet problem 4.9. Te Fredolm alternative (4.26) Lu = f in, u = on, were is a smoot, bounded open set, and Lu = i (a ij j u) + i (b i u) + cu. If u = v = on, Green s formula implies tat (Lu)v dx = u (L v) dx, were te formal adjoint L of L is defined by L v = i (a ij j v) b i i v + cv. It follows tat if u is a smoot solution of (4.26) and v is a smoot solution of te omogeneous adjoint problem, ten L v = in, v = on, fv dx = (Lu)v dx = ul v dx =. Tus, a necessary condition for (4.26) to be solvable is tat f is ortogonal wit respect to te L 2 ()-inner product to every solution of te omogeneous adjoint problem. For bounded domains, we will use te compactness of te resolvent to prove tat tis condition is necessary and sufficient for te existence of a weak solution of (4.26) were f L 2 (). Moreover, te solution is unique if and only if a solution exists for every f L 2 (). Tis result is a consequence of te fact tat if K is compact, ten te operator I+σK is a Fredolm operator wit index zero on L 2 () for any σ R, and terefore satisfies te Fredolm alternative (see Section 4.B.2). Tus, if K = (L + µi) 1 is compact, te inverse elliptic operator L λi also satisfies te Fredolm alternative. Teorem Suppose tat is a bounded open set in R n and L is a uniformly elliptic operator of te form (4.14) wose coefficients satisfy (4.15). Let L be te adjoint operator (4.2) and λ R. Ten one of te following two alternatives olds. (1) Te only weak solution of te equation L v λv = is v =. For every f L 2 () tere is a unique weak solution u H 1 () of te equation Lu λu = f. In particular, te only solution of Lu λu = is u =.

17 4.9. THE FREDHOLM ALTERNATIVE 13 (2) Te equation L v λv = as a nonzero weak solution v. Te solution spaces of Lu λu = and L v λv = are finite-dimensional and ave te same dimension. For f L 2 (), te equation Lu λu = f as a weak solution u H 1 () if and only if (f, v) = for every v H 1 () suc tat L v λv =, and if a solution exists it is not unique. Proof. Since K = (L + µi) 1 is a compact operator on L 2 (), te Fredolm alternative olds for te equation (4.27) u + σku = g u, g L 2 () for any σ R. Let us consider te two alternatives separately. First, suppose tat te only solution of v + σk v = is v =, wic implies tat te only solution of L v + (µ + σ)v = is v =. Ten te Fredolm alterative for I +σk implies tat (4.27) as a unique solution u L 2 () for every g L 2 (). In particular, for any g ran K, tere exists a unique solution u L 2 (), and te equation implies tat u ran K. Hence, we may apply L + µi to (4.27), and conclude tat for every f = (L + µi)g L 2 (), tere is a unique solution u ran K H 1 () of te equation (4.28) Lu + (µ + σ)u = f. Taking σ = (λ + µ), we get part (1) of te Fredolm alternative for L. Second, suppose tat v + σk v = as a finite-dimensional subspace of solutions v L 2 (). It follows tat v ran K (clearly, σ in tis case) and L v + (µ + σ)v =. By te Fredolm alternative, te equation u + σku = as a finite-dimensional subspace of solutions of te same dimension, and ence so does Lu + (µ + σ)u =. Equation (4.27) is solvable for u L 2 () given g ran K if and only if (4.29) (v, g) L 2 = for all v L 2 () suc tat v + σk v =, and ten u ran K. It follows tat te condition (4.29) wit g = Kf is necessary and sufficient for te solvability of (4.28) given f L 2 (). Since (v, g) L 2 = (v, Kf) L 2 = (K v, f) L 2 = 1 σ (v, f) L 2 and v + σk v = if and only if L v + (µ + σ)v =, we conclude tat (4.28) is solvable for u if and only if f L 2 () satisfies (v, f) L 2 = for all v ran K suc tat L v + (µ + σ)v =. Taking σ = (λ + µ), we get alternative (2) for L. Elliptic operators on a Riemannian manifold may ave nonzero Fredolm index. Te Atiya-Singer index teorem (1968) relates te Fredolm index of suc operators wit a topological index of te manifold.

18 14 4. ELLIPTIC PDES 4.1. Te spectrum of a self-adjoint elliptic operator Suppose tat L is a symmetric, uniformly elliptic operator of te form (4.3) Lu = i (a ij j u) + cu were a ij = a ji and a ij, c L (). Te associated symmetric bilinear form is given by a(u, v) = a : H 1 () H 1 () R a ij i u j u + cuv dx. Te resolvent K = (L + µi) 1 is a compact self-adjoint operator on L 2 () for sufficiently large µ. Terefore its eigenvalues are real and its eigenfunctions provide an ortonormal basis of L 2 (). Since L as te same eigenfunctions as K, we get te corresponding result for L. Teorem Te operator L as an increasing sequence of real eigenvalues of finite multiplicity λ 1 < λ 2 λ 3 λ n... suc tat λ n. Tere is an ortonormal basis {φ n : n N} of L 2 () consisting of eigenfunctions functions φ n H 1 () suc tat Lφ n = λ n φ n. Proof. If Kφ = for any φ L 2 (), ten applying L + µi to te equation we find tat φ =, so is not an eigenvalue of K. If Kφ = κφ, for φ L 2 () and κ, ten φ ran K and ( ) 1 Lφ = κ µ φ, so φ is an eigenfunction of L wit eigenvalue λ = 1/κ µ. From Gårding s inequality (4.21) wit u = φ, and te fact tat a(φ, φ) = λ φ 2 L, we get 2 C 1 φ 2 H (λ + γ) φ 2 1 L 2. It follows tat λ > γ, so te eigenvalues of L are bounded from below, and at most a finite number are negative. Te spectral teorem for te compact selfadjoint operator K ten implies te result. Te boundedness of te domain is essential ere, oterwise K need not be compact, and te spectrum of L need not consist only of eigenvalues. Example Suppose tat = R n and L =. Let K = ( + I) 1. Ten, from Example 4.12, K : L 2 (R n ) L 2 (R n ). Te range of K is H 2 (R n ). Tis operator is bounded but not compact. For example, if φ Cc (R n ) is any nonzero function and {a j } is a sequence in R n suc tat a j as j, ten te sequence {φ j } defined by φ j (x) = φ(x a j ) is bounded in L 2 (R n ) but {Kφ j } as no convergent subsequence. In tis example, K as continuous spectrum [, 1] on L 2 (R n ) and no eigenvalues. Correspondingly, as te purely continuous spectrum [, ).

19 4.11. INTERIOR REGULARITY 15 Finally, let us briefly consider te Fredolm alternative for a self-adjoint elliptic equation from te perspective of tis spectral teory. Te equation (4.31) Lu λu = f may be solved by expansion wit respect to te eigenfunctions of L. Suppose tat {φ n : n N} is an ortonormal basis of L 2 () suc tat Lφ n = λ n φ n, were te eigenvalues λ n are increasing and repeated according to teir multiplicity. We get te following alternatives, were all series converge in L 2 (): (1) If λ λ n for any n N, ten (4.31) as te unique solution u = n=1 (f, φ n ) λ n λ φ n for every f L 2 (); (2) If λ = λ M for for some M N and λ n = λ M for M n N, ten (4.31) as a solution u H 1 () if and only if f L 2 () satisfies In tat case, te solutions are (f, φ n ) = for M n N. u = λ n λ (f, φ n ) λ n λ φ n + N c n φ n n=m were {c M,..., c N } are arbitrary real constants Interior regularity Rougly speaking, solutions of elliptic PDEs are as smoot as te data allows. For boundary value problems, it is convenient to consider te regularity of te solution in te interior of te domain and near te boundary separately. We begin by studying te interior regularity of solutions. We follow closely te presentation in [5]. To motivate te regularity teory, consider te following simple a priori estimate for te Laplacian. Suppose tat u Cc (R n ). Ten, integrating by parts twice, we get ( u) 2 dx = = = ( 2 iiu ) ( 2 jju ) dx ( 3 iiju ) ( j u) dx ( 2 iju ) ( 2 iju ) dx = D 2 u 2 dx. Hence, if u = f, ten D 2 u L2 = f 2 L 2. Tus, we can control te L 2 -norm of all second derivatives of u by te L 2 -norm of te Laplacian of u. Tis estimate suggests tat we sould ave u Hloc 2 if f, u L 2, as is in fact true. Te above computation is, owever, not justified for

20 16 4. ELLIPTIC PDES weak solutions tat belong to H 1 ; as far as we know from te previous existence teory, suc solutions may not even possess second-order weak derivatives. We will consider a PDE (4.32) Lu = f in were is an open set in R n, f L 2 (), and L is a uniformly elliptic of te form (4.33) Lu = i (a ij j u). It is straigtforward to extend te proof of te regularity teorem to uniformly elliptic operators tat contain lower-order terms [5]. A function u H 1 () is a weak solution of (4.32) (4.33) if (4.34) a(u, v) = (f, v) for all v H 1 (), were te bilinear form a is given by (4.35) a(u, v) = a ij i u j v dx. We do not impose any boundary condition on u, for example by requiring tat u H 1 (), so te interior regularity teorem applies to any weak solution of (4.32). Before stating te teorem, we illustrate te idea of te proof wit a furter a priori estimate. To obtain a local estimate for D 2 u on a subdomain, we introduce a cut-off function η Cc () suc tat η 1 and η = 1 on. We take as a test function (4.36) v = k η 2 k u. Note tat v is given by a positive-definite, symmetric operator acting on u of a similar form to L, wic leads to te positivity of te resulting estimate for D k u. Multiplying (4.32) by v and integrating over, we get (Lu, v) = (f, v). Two integrations by parts imply tat (Lu, v) = j (a ij i u) ( k η 2 k u ) dx were F = = = { η 2 ( k a ij ) ( i u) ( j k u) k (a ij i u) ( j η 2 k u ) dx η 2 a ij ( i k u) ( j k u) dx + F [ + 2η j η a ij ( i k u) ( k u) + ( k a ij ) ( i u) ( k u)] } dx.

21 4.11. INTERIOR REGULARITY 17 Te term F is linear in te second derivatives of u. We use te uniform ellipticity of L to get θ D k u 2 dx η 2 a ij ( i k u) ( j k u) dx = (f, v) F, and a Caucy inequality wit ɛ to absorb te linear terms in second derivatives on te rigt-and side into te quadratic terms on te left-and side. Tis results in an estimate of te form ) D k u 2 L 2 ( ) (f C 2 + u 2 H 1 (). Te proof of regularity is entirely analogous, wit te derivatives in te test function (4.36) replaced by difference quotients (see Section 4.C). We obtain an L 2 ( )- bound for te difference quotients D k u tat is uniform in, wic implies tat u H 2 ( ). Teorem Suppose tat is an open set in R n. Assume tat a ij C 1 () and f L 2 (). If u H 1 () is a weak solution of (4.32) (4.33), ten u H 2 ( ) for every. Furtermore, (4.37) u H2 ( ) C ( ) f L 2 () + u L 2 () were te constant C depends only on n,, and a ij. Proof. Coose a cut-off function η Cc () suc tat η 1 and η = 1 on. We use te compactly supported test function v = D ( k η 2 Dku ) H 1 () in te definition (4.34) (4.35) for weak solutions. (As in (4.36), v is given by a positive self-adjoint operator acting on u.) Tis implies tat (4.38) a ij ( i u) D k ( j η 2 Dku ) dx = fd ( k η 2 Dku ) dx. Performing a discrete integration by parts and using te product rule, we may write te left-and side of (4.38) as (4.39) a ij ( i u) D k ( j η 2 Dku ) dx = = were, wit a ij (x) = a ij(x + e k ), { F = η 2 ( Dka ) ij ( i u) ( Dk j u ) (4.4) D k (a ij i u) j ( η 2 D ku ) dx η 2 a ij ( D k i u ) ( D k j u ) dx + F, [ + 2η j η a ( ij D k i u ) ( Dku ) + ( Dka ) ij ( i u) ( Dku )]} dx.

22 18 4. ELLIPTIC PDES Using te uniform ellipticity of L in (4.16), we estimate θ η 2 D k Du 2 dx Using (4.38) (4.39) and tis inequality, we find tat (4.41) θ η 2 D k Du 2 dx η 2 a ij ( D k i u ) ( D k j u ) dx. fd ( k η 2 Dku ) dx F. By te Caucy-Scwartz inequality, fd ( k η 2 Dku ) dx f ( L 2 () D k η 2 Dku ) L 2 (). Since spt η, Proposition 4.52 implies tat for sufficiently small, ( D k η 2 Dku ) ( k L2 () η 2 Dku ) L2 () A similar estimate of F in (4.4) gives ( F C Du L2 () Using tese results in (4.41), we find tat (4.42) η 2 k D ku L2 () + 2η ( k η) D ku L2 () η k D ku L2 () + C Du L 2 (). ηd kdu ) + L 2 () Du 2 L 2 (). θ ηd k Du 2 C ( f L 2 () L 2 () ηd k Du + f L2 () L 2 () Du L 2 () + Du L 2 () ηd k Du ) + L2 () Du 2 L 2 (). By Caucy s inequality wit ɛ, we ave f L 2 () ηd k Du L 2 () ɛ ηd k Du 2 L 2 () + 1 4ɛ f 2 L 2 (), Du L 2 () ηd k Du L2 () ɛ ηd k Du 2 L 2 () + 1 4ɛ Du 2 L 2 (). Hence, coosing ɛ so tat 4Cɛ = θ, and using te result in (4.42) we get tat θ ηd 4 k Du ( ) 2 L 2 () C f 2 L 2 () + Du 2 L 2 (). Tus, since η = 1 on, (4.43) D k Du ( ) 2 L 2 ( ) C f 2 L 2 () + Du 2 L 2 () were te constant C depends on,, a ij, but is independent of, u, f.

23 4.12. BOUNDARY REGULARITY 19 We can furter estimate Du in terms of u by taking v = u in (4.34) (4.35) and using te uniform ellipticity of L to get θ Du 2 dx a ij i u j u fu dx f L 2 () u L 2 () 1 ( ) f 2 L 2 2 () + u 2 L 2 (). Using tis result in (4.43), we get tat D kdu ( ) 2 L 2 ( ) C f 2 L 2 () + u 2 L 2 (). Teorem 4.53 teorem now implies tat te weak second derivatives of u exist and belong to L 2 (). Furtermore, te H 2 -norm of u satisfies (4.37). If u H 2 loc () and f L2 (), ten te equation Lu = f relating te weak derivatives of u and f olds pointwise a.e.; suc solutions are often called strong solutions, to distinguis tem from weak solutions wic may not possess weak second order derivatives and classical solutions wic possess continuous second order derivatives. Te repeated application of tese estimates leads to iger interior regularity. Teorem Suppose tat a ij C k+1 () and f H k (). If u H 1 () is a weak solution of (4.32) (4.33), ten u H k+2 ( ) for every. Furtermore, u H k+2 ( ) C ( ) f H k () + u L2 () were te constant C depends only on n, k,, and a ij. See [5] for a detailed proof. Note tat if te above conditions old wit k > n/2, ten f C() and u C 2 (), so u is a classical solution of te PDE Lu = f. Furtermore, if f and a ij are smoot ten so is te solution. Corollary If a ij, f C () and u H 1 () is a weak solution of (4.32) (4.33), ten u C () Proof. If, ten f H k ( ) for every k N, so by Teorem (4.28) u H k+2 loc ( ) for every k N, and by te Sobolev imbedding teorem u C ( ). Since tis olds for every open set, we ave u C () Boundary regularity To study te regularity of solutions near te boundary, we localize te problem to a neigborood of a boundary point by use of a partition of unity: We decompose te solution into a sum of functions tat are compactly supported in te sets of a suitable open cover of te domain and estimate eac function in te sum separately. Assuming, as in Section 1.1, tat te boundary is at least C 1, we may flatten te boundary in a neigborood U by a diffeomorpism ϕ : U V tat maps U to an upper alf space V = B 1 () {y n > }. If ϕ 1 = ψ and x = ψ(y), ten by a

24 11 4. ELLIPTIC PDES cange of variables (c.f. Teorem 1.38 and Proposition 3.2) te weak formulation (4.32) (4.33) on U becomes ũ ṽ ã ij dy = fṽ dy for all functions ṽ H 1 (V ), y i y j V V were ũ H 1 (V ). Here, ũ = u ψ, ṽ = v ψ, and ( ) ( ) ϕi ϕj ã ij = det Dψ a pq ψ ψ, f = det Dψ f ψ. x p x q p,q=1 Te matrix ã ij satisfies te uniform ellipticity condition if a pq does. To see tis, we define ζ = (Dϕ t ) ξ, or ϕ i ζ p = ξ i. x p Ten, since Dϕ and Dψ = Dϕ 1 are invertible and bounded away from zero, we ave for some constant C > tat ã ij ξ i ξ j = det Dψ a pq ζ p ζ q det Dψ θ ζ 2 Cθ ξ 2. i,j p,q=1 Tus, we obtain a problem of te same form as before after te cange of variables. Note tat we must require tat te boundary is C 2 to ensure tat ã ij is C 1. It is important to recognize tat in canging variables for weak solutions, we need to verify te cange of variables for te weak formulation directly and not for te original PDE. A transformation tat is valid for smoot solutions of a PDE is not always valid for weak solutions, wic may lack sufficient smootness to justify te transformation. We now state a boundary regularity teorem. Unlike te interior regularity teorem, we impose a boundary condition u H 1 () on te solution, and we require tat te boundary of te domain is smoot. A solution of an elliptic PDE wit smoot coefficients and smoot rigt-and side is smoot in te interior of its domain of definition, watever its beavior near te boundary; but we cannot expect to obtain smootness up to te boundary witout imposing a smoot boundary condition on te solution and requiring tat te boundary is smoot. Teorem 4.3. Suppose tat is a bounded open set in R n wit C 2 -boundary. Assume tat a ij C 1 () and f L 2 (). If u H 1 () is a weak solution of (4.32) (4.33), ten u H 2 (), and u H2 () C ( ) f L2 () + u L2 () were te constant C depends only on n, and a ij. Proof. By use of a partition of unity and a flattening of te boundary, it is sufficient to prove te result for an upper alf space = {(x 1,..., x n ) : x n > } space and functions u, f : R tat are compactly supported in B 1 (). Let η Cc (R n ) be a cut-off function suc tat η 1 and η = 1 on B 1 (). We will estimate te tangential and normal difference quotients of Du separately. First consider a test function tat depends on tangential differences, v = D k η2 D ku for k = 1, 2,..., n 1.

25 4.12. BOUNDARY REGULARITY 111 Since te trace of u is zero on, te trace of v on is zero and, by Teorem 3.42, v H 1 (). Tus we may use v in te definition of weak solution to get (4.38). Exactly te same argument as te one in te proof of Teorem 4.27 gives (4.43). It follows from Teorem 4.53 tat te weak derivatives k i u exist and satisfy ) (4.44) k Du L 2 () ( f C 2 L 2 () + u 2 L 2 () for k = 1, 2,..., n 1. Te only derivative tat remains is te second-order normal derivative nu, 2 wic we can estimate from te equation. Using (4.32) (4.33), we ave for φ Cc () tat a nn ( n u) ( n φ) dx = a ij ( i u) ( j φ) dx + fφ dx were denotes te sum over 1 i, j n wit te term i = j = n omitted. Since a ij C 1 () and i u is weakly differentiable wit respect to x j unless i = j = n we get, using Proposition 3.2, tat a nn ( n u) ( n φ) dx = { j [a ij ( i u)] + f} φ dx for every φ Cc (). It follows tat a nn ( n u) is weakly differentiable wit respect to x n, and { } n [a nn ( n u)] = j [a ij ( i u)] + f L 2 (). From te uniform ellipticity condition (4.16) wit ξ = e n, we ave a nn θ. Hence, by Proposition 3.2, n u = 1 a nn n u a nn is weakly differentiable wit respect to x n wit derivative 2 nnu = 1 a nn n [a nn n u] + n ( 1 a nn ) a nn n u L 2 (). Furtermore, using (4.44) we get an estimate of te same form for nnu 2 2 L 2 (), so tat D 2 u ( ) C L f 2 2 () L 2 () + u 2 L 2 () Te repeated application of tese estimates leads to iger-order regularity. Teorem Suppose tat is a bounded open set in R n wit C k+2 - boundary. Assume tat a ij C k+1 () and f H k (). If u H 1 () is a weak solution of (4.32) (4.33), ten u H k+2 () and u H k+2 () C ( ) f Hk () + u L 2 () were te constant C depends only on n, k,, and a ij. Sobolev imbedding ten yields te following result. Corollary Suppose tat is a bounded open set in R n wit C boundary. If a ij, f C () and u H 1 () is a weak solution of (4.32) (4.33), ten u C ()

26 ELLIPTIC PDES Some furter perspectives Te above results give an existence and L 2 -regularity teory for second-order, uniformly elliptic PDEs in divergence form. Tis teory is based on te simple a priori energy estimate for Du L 2 tat we obtain by multiplying te equation Lu = f by u, or some derivative of u, and integrating te result by parts. Tis teory is a fundamental one, but tere is a bewildering variety of approaces to te existence and regularity of solutions of elliptic PDEs. In an attempt to put te above analysis in a broader context, we briefly list some of tese approaces and oter important results, witout any claim to completeness. Many of tese topics are discussed furter in te references [5, 1, 12]. L p -teory: If 1 < p <, tere is a similar regularity result tat solutions of Lu = f satisfy u W 2,p if f L p. Te derivation is not as simple wen p 2, owever, and requires te use of more sopisticated tools from real analysis (suc as Calderón-Zygmund operators in armonic analysis). Scauder teory: Te Scauder teory provides Hölder-estimates similar to tose derived in Section for Laplace s equation, and a corresponding existence teory of solutions u C 2,α of Lu = f if f C,α and L as Hölder continuous coefficients. General linear elliptic PDEs are treated by regarding tem as perturbations of constant coefficient PDEs, an approac tat works because tere is no loss of derivatives in te estimates of te solution. Te Hölder estimates were originally obtained by te use of potential teory, but oter ways to obtain tem are now known; for example, by te use of Campanato spaces, wic provide Hölder norms in terms of suitable integrals tat are easier to estimate directly. Perron s metod: Perron (1923) sowed tat solutions of te Diriclet problem for Laplace s equation can be obtained as te infimum of superarmonic functions or te supremum of subarmonic functions, togeter wit te use of barrier functions to prove tat, under suitable assumptions on te boundary, te solution attains te prescribed boundary values. Tis metod is based on maximum principle estimates. Boundary integral metods: By te use of Green s functions, one can often reduce a linear elliptic BVP to an integral equation on te boundary, and ten use te teory of integral equations to study te existence and regularity of solutions. Tese metods also provide efficient numerical scemes because of te lower dimensionality of te boundary. Pseudo-differential operators: Te Fourier transform provides an effective metod for solving linear PDEs wit constant coefficients. Te teory of pseudo-differential and Fourier-integral operators is a powerful extension of tis metod tat applies to general linear PDEs wit variable coefficients, and elliptic PDEs in particular. It is, owever, less well-suited to te analysis of nonlinear PDEs. Variational metods: Many elliptic PDEs especially tose in divergence form arise as Euler-Lagrange equations for variational principles. Existence of weak solutions can often be sown by use of te direct metod of te calculus of variations, after wic one studies te regularity of a minimizer (or, in some cases, a critical point). Di Giorgi-Nas-Moser: Te work of Di Giorgi (1957), Nas (1958), and Moser (196) sowed tat weak solutions of a second order elliptic PDE

27 4.13. SOME FURTHER PERSPECTIVES 113 in divergence form wit bounded (L ) coefficients are Hölder continuous (C,α ). Tis was te key step in developing a regularity teory for minimizers of nonlinear variational principles wit elliptic Euler-Lagrange equations. Moser also obtained a Harnack inequality for weak solutions. Fully nonlinear equations: Krylov and Safonov (1979) obtained a Harnack inequality for second order elliptic equations in nondivergence form. Tis allowed te development of a regularity teory for fully nonlinear elliptic equations (e.g. second-order equations for u tat depend nonlinearly on D 2 u). Crandall and Lions (1983) introduced te notion of viscosity solutions wic despite te name uses te maximum principle and is based on a comparison wit appropriate sub and super solutions Tis teory applies to fully nonlinear elliptic PDEs, altoug it is mainly restricted to scalar equations. Degree teory: Topological metods based on te Leray-Scauder degree of a mapping on a Banac space can be used to prove existence of solutions of various nonlinear elliptic problems (see e.g. L. Nirenberg, Topics in Nonlinear Functional Analysis). Tese metods can provide global existence results for large solutions, but often do not give muc detailed analytical information about te solutions. Heat flow metods: Parabolic PDEs, suc as u t + Lu = f, are closely connected wit te associated elliptic PDEs for stationary solutions, suc as Lu = f. One may use tis connection to obtain solutions of an elliptic PDE as te limit as t of solutions of te associated parabolic PDE. For example, Hamilton (1981) introduced te Ricci flow on a manifold, in wic te metric approaces a Ricci-flat metric as t, as a means to understand te topological classification of smoot manifolds, and Perelman (23) used tis approac to prove te Poincaré conjecture (tat every simply connected, tree-dimensional, compact manifold witout boundary is omeomorpic to a tree-dimensional spere) and, more generally, te geometrization conjecture of Turston.