Vorlesung: Semi-linear Elliptic PDEs (WS 2015/16)



Content of the lecture:

[BS] M. Badiale, E. Serra (2011), Semilinear Elliptic Equations for Beginners, Springer (Universitext), 2011.

In-official, non-corrected, and not necessary up-to-date TeX'ed version of Marcel Schaub's notes from the lecture can be found here.



14 October:

Introduction, practical Information (see main page).

Various things on the content of the course, and the book by Badiale & Serra. Discussion of 'The Variational Approach'.

[BS] Preface (pp v-vii), Section 1.1 (pp 1-4).

Chapter 1: Notation, repetition (Sobolev spaces, linear theory).

Various notation; various function spaces.

[BS] Section 1.2: 1.2.1 (p 5).

21 October:

Sobolev spaces (H^1 and H^1_0), Sobolev embeddings (continuous, and compact), dominated convergence, Poincaré's inequality, Riesz' representation theorem, Banach-Alaoglu.
Important consequence:
Proposition: A sequence in H^1_0 with uniformly bounded Dirichlet-integrals has a subsequence which converges weakly in H^1_0, strongly in L^p for p in [1, 2N/(N-2)), and pointwise a.e. It also has an L^p-dominant.

2nd order linear PDO, divergence and non-divergence form, uniform ellipticity. For variational approach: NO 1st order terms. Main example for principle part (will do only): Laplacian. Discussion.
Definition of classical and weak solutions to (homogeneous) Dirichlet B(oundary) V(alue) P(roblem) (BVP). Discussion and remarks..

Description of Variational Approach to solving the (linear) Dirichlet BVP: Write equation (in weak form) as J'(u)=0 for a suitable (nonlinear) functional J on H^1_0. Idea: Extend to (certain) nonlinear eq's.

Spectral properties of elliptic operators (on domain, with Dirichlet B(oundary) C(ondition) (BC)):
For L = - Delta + q, q bounded (on open & bounded domain Omega).
Theorem: On eigenvalues, eigenvectors of L. Discussion and remarks..
Theorem: Variational characterization of Principle Eigenvalue. Discussion and remarks.

[BS] Sections 1.2.1, 1.2.2, 1.2.3, 1.2.4, (pp 5-11), 1.4 (pp 22-25), 1.7 (pp 31-33; to be continued).
See also PDE2 (Lecture and Exercises and Evans' book).

04 November:

Continuation of remarks from last time: Expansion in ONB of eigenfunctions in L^2 as well as in H^1_0.

Fredholm's Alternative for the existence of weak solutions to inhomogeneous _linear_ equations (with spectral parameter).

Chapter 2: Differential calculus in Banach spaces, convexity, and first results.

Notation/definition: Functionals on open subsets of Banach spaces.
Definition: (Fréchet) differentiable at u in U, on U, (Fréchet) derivative, C^1 on U. Remarks.
Definition: The gradient of a differentiable functional on a Hilbert space.
Propostion: 'Natural computation rules'.
Definition: Gâteaux differentiable at u in U, on U, Gâteaux derivative.

Motivation: Gâteaux weaker than Fréchet, but (often) easier to prove - and have:

Proposition: If functional is Gâteaux diff on U, and the Gâteaux derivative is continuous at u_0 in U, then the functional is Fréchet diff at u_0, and the two differentials agree.
(NO PROOF! - See [AP] Ambrosetti and Prodi, A Primer on Nonlinear Analysis, CUP).
Remarks.

Definition: Critical point, critical point at level c, critical level, Euler-Lagrange (E-L) equation.

Abstract examples of differentiable functionals: Constant functional, linear functional, quadratic functionals (from quadratic forms), symmetric quadratic functionals (from symmetric quadratic forms), in Banach and in Hilbert space; derivative of square of norm in Hilbert space and its gradient; derivative of norm (away from 0), and its gradient; quotient functionals. Remarks.

Concrete examples of differentiable quadratic functionals, in L^2, H^1_0, H^1 (continued next time).

[BS] Sections 1.7 (pp 32-34), 1.3, 1.3.1, 1.3.2 (pp 11-17).

11 November:

Concrete examples of differentiable _quadratic_ functionals, in L^2, H^1_0, H^1 (continued from last time).

Concrete example of differentiable _quadratic_ functional in H^1 (with bounded matrix A and bounded q).

Concrete example of 'quotient functional' (using the functional above; generalising the 'Rayleigh quotient').

Concrete example of differentiable _quadratic_ functional, given by the bilinear form of integrating against q with q in L^{N/2}(Omega), for Omega open and _bounded_ subset of R^N.

Proposition: For Omega open and _bounded_ subset of R^N, and continuous f : R -> R satisfying (f)_1:
|f(t)| <= a + b|t|^{2^*-1} (with 2^* = 2N/(N-2), critical Sobolev constant), let F(t) = int_0^t f(s) ds, and define J(u) = int F(u(x)) dx, u in H^1. Then J is (well-defined and) Fréchet diff on H^1_0 (and, if boundary Omega C^1, on H^1) and
J'(u)v = int f(u(x))v(x) dx. (End of proof next time.)

[BS] Section 1.3.2 (pp 16-20), p 30: line 12 (and Exercise 4 p 36).
The Proposition above is [BS] 1.3.20; the condition (f)_1 in Lecture is (1.9) in [BS].

18 November:

End of proof from last time (see above).

Proposition: For Omega open and _unbounded_ subset of R^N, and continuous f : R -> R satisfying
|f(t)| <= a|t| + b|t|^{2^*-1} (with 2^* = 2N/(N-2), critical Sobolev constant), let F(t) = int_0^t f(s) ds, and define J(u) = int F(u(x)) dx, u in H^1. Then J is (well-defined and) Fréchet diff on H^1_0 (and, if boundary Omega C^1, on H^1) and
J'(u)v = int f(u(x))v(x) dx. (NO proof!)

Definition: Weak solution to (semi-linear) Dirichlet BVP (with f satisfying (f)_1).

Proposition: u in H^1_0 is a weak solution iff it is a critical of the corresponding (differentiable!) functional.

Remark: On the condition (f)_1 (which will _always_ be assumed).

Convex functionals: Discussion of min/max.

Lemma: If a differentiable functional has a (local) minimum/maximum point, then that is a critical point.

Definition: Convex, strict convex, concave, strict concave.

Proposition: A strictly convex functional has at _most_ one minimum point.

Proposition: A strictly convex and differentiable functional have at _most_ one critical point.

Proposition: Let I:X -> R be differentiable functional, with
[I'(u)-I'(v)](u-v) >=0 for all u,v in X.
Then I is convex. Strict inequality gives strict convexity.

Discussion of 'lack of minima for convex functions/functionals'.

Definition: Coercive functional.

Definition: Minimising sequence for a real valued map.

[BS] Sections 1.3.2 (pp 18-22); 1.4 (pp 22-25); 1.5 (pp 25-28; to be continued).

25 November:

Remark: A minimising sequence _always_ exist.

Lemma: A minimising sequence for a coercive functional J on a normed space X is bounded.

Discussion: of the 'Direct Mehod in the Calculus of Variations': Tactics and problems.

Definition: Lower/upper semi-continuity; lower/upper _sequential_ semi-continuity (of real-valued maps on topological space). Remarks, discussion.

Proposition: A _convex_ subset K of a Banach space X is norm-closed iff it is weakly closed. Remarks, discussion.

Lemma: A _convex_ and (norm) lower semi-cont functional J on a convex and (norm) closed subset K of a Banach space X is _weakly_ lower semi-cont. In particular, the norm of a Banach space X is weakly lower semi-cont. Remarks and discussions.

Theorem (Weierstrass): A weakly sequential lower semi-cont real functional J on a convex and (norm) closed subset K of a _reflexiv_ Banach space X is bounded from below and attains its minimum (on K) IF either (1) K is _bounded_ or (2) J is _coercive_ on K.

Theorem: A (norm) lower semi-cont real functional J on a convex and (norm) closed subset K of a _reflexiv_ Banach space X is bounded from below and attains its minimum (on K) IF either (1) K is _bounded_ or (2) J is _coercive_ on K. If J is _strictly_ convex, this minimum is unique. Remarks, discussions.

Corollary: A (norm) cont, convex, and coercive real functional on a reflexive Banach space has a global minimum point.
Discussion: Tactic to show _existence_ of weak solutions to certain PDE's as critical points of certain functionals, by showing this functional has a minimum point.

Theorem: First result on existence of _unique_ weak solutions to BVP for (q bounded, q>=0):
- Δu + qu = f(u) + h(x)
for _all_ h in L^2, IF f is cont, satisfies (f)_1, and also
(f)_2: f(t)t <= 0
and
(f)_3: f is non-increasing. (Proof next time).

[BS] Sections 1.5 (pp 26-28); 1.6 (pp 28-31; to be continued).
See also [K] Kavian, Introduction à la théorie des points critiques, Springer (1993) (pp 135-136).

02 December:

Theorem (stated last time): First result on existence of _unique_ weak solutions to BVP for (q bounded, q>=0):
- Δu + qu = f(u) + h(x)
for _all_ h in L^2, IF f is cont, satisfies (f)_1, and also
(f)_2: f(t)t <= 0
and
(f)_3: f is non-increasing.

Remarks, discussions: Generalisations.

Corollary: Existence of _unique_ weak solutions to (linear!) BVP for (q bounded, q>=0):
- Δu + qu = h
for _all_ h in L^2.

Corollary: Existence of _unique_ weak solutions (for all h in L^2) to (nonlinear!) BVP for
- Δu +|u|^{p-2}u = h
IF p in (1,2^*]. In particular, - Δu +|u|^{p-2}u = 0 ONLY has trivial solution u=0.

Proof of theorem above.

Remark: (a) (f)_2 gives coercivity, (f)_3 gives strict convexity.
(b) If f(0)=0 then u=0 is (weak) solution to (BVP of) - Δu + qu = f(u).
(c) If (f)_2 holds, then u=0 is _unique_ solution of this.

Remark: When do _not_ have convexity of J, may still prove weakly seq. l.s.c. _directly_ (without convexity) - and prove also coercivity, to conclude existence of minimiser (via Weierstrass).

Proposition: (a) If f cont satisfies
ε-(f)_1: |f(t)| <= a + b|t|^{p-1} for p in (0,2^{*})
then K(u)=int F(u) dx, with F(t)=int_0^t f(s)ds, is weakly seq cont on H^1_0.
(b) If f cont satisfies
(f)_1: |f(t)| <= a + b|t|^{2^*-1}
_and_
(f)_2: f(t)t <=0,
then -K(u) is weakly seq _lower_ s.c. on H^1_0.
(End of proof next time).

[BS] 1.6 (pp 29-31); 2.1 (pp 39-45; to be continued).
By now, also read (!) [BS] Sections 1.8 (pp 35-37) and 1.9 (p 37).

09 December:

End of proof from last time (see above).

Remark: (f)_1 ensures K(u) is Fréchet-diff, hence _norm_ cont. Problem is to prove _weakly_ seq. l.s.c.

Proposition: Weakly seq l.s.c. of J with non-linearity given by (cont) f if
(1) ε-(f)_1: |f(t)| <= a + b|t|^{p-1} for p in (0,2^{*})
OR
(2) (f)_1: |f(t)| <= a + b|t|^{2^*-1} _and_ (f)_2: f(t)t <=0.

Proposition: Coercivity of J with non-linearity given by (cont) f if
(f)_4 |f(t)| <= a + b|t|, b in (0,λ_1).
(λ_1=λ_1(-Δ + q) first eigenvalue; we talk about 'linear growth').

Remark: (f)_4 holds if f has 'sublinear growth' or is bounded (which might be more easy to check than (f)_4).
IF (f)_4 holds with b>λ_1, there are equations with _no_ weak solutions (hence, J is not coercive, see next result).

Theorem: Existence of _a_ weak solution (for all h in L^2) to BVP for (q bounded, q>=0):
- Δu + qu = f(u) + h(x)
IF (cont) f satisfies (f)_4.

Remark: Solution need not be unique; resume of 'tactic'.

Corollary: Existence of _a_ weak solution (for all h in L^2) to BVP for (q bounded, q>=0):
- Δu + qu = f(u) + h(x)
IF (cont) f has sublinear growth.

Theorem: Existence of _a_ weak solution (for all h in L^2) to BVP for (q bounded, q>=0):
- Δu + qu = f(u) + h(x)
IF (cont) f satisfies (f)_1 and (f)_2.

Remark: Here, allow superlinear growth (including critical growth) but have a sign condition: ensures coercivity.

Remark: On equations with _only_ trivial (u=0) solution.

Theorem: Existence of _a_ weak _non-trivial_ solution to BVP for (q bounded, q>=0):
- Δu + qu = f(u) + h(x)
IF (cont) f satisfies (f)_4 AND
(f)_5 f(t) >= β t for t in [0,δ] (some δ>0, some β>λ_1.

Brief (!) discussion of rest of Chapter 2 + Chapter 3 in [BS].

[BS] 2.1 (pp 39-45).

By now, also read (!) [BS] Section 2.8 (pp 95-96: On Section 2.1).

16 December:

Chapter 3: Minimax Methods.

Ex: Nonlinear BVP for (q bounded, q>=0)
- Δu + qu = |u|^{p-2}u
for p in (2,2^*) ('superlinear, subcritical'): u is weak sol iff u is critical point of J(u)= (1/2)||u||_q^2 - (1/p)|u|_p^p. J is not bounded from below. u=0 is critical point of J. By Sobolev: it is a local minimum (at level 0). There exists v with J(v) < 0. In 1 dimension (d=1): This would ensure existence of _second_ (non-trivial) critical point (ex: f(x) = x^2 - x^4).

Ex: If g : R -> R has two local minima, it must have a third critical point between them.

Ex: These things _fails_ in higher dimension (d=2 enough): G(x,y)=x^2(1+y)^3+7y^2 has a _single_ critical point which is local but not global minimum. F(x,y)=(x^2y-x-1)^2+(x^2-1)^2 has _exactly_ two critical points, both local minima. In fact: For all n there exists polynomial P_n(x,y) with _exactly_ n local minima of which none are global minima.

Definition: Sublevel sets (sublevels) J^a of functional J; u a point at level c (of J).

Ex: Change of sublevels for d=1 (x^2-x^4, x^3-3x, a function g with no critical points at levels in [a,b]; x*exp(1-x)).

Definition: Palais-Smale (PS) sequence for J, and for J at level c.

Ex: PS-seq's for x*exp(1-x), at level 0 and level 1.

Remarks and discussions (f.ex., in Hilbert space, will identify J'(u) with grad J(u)).

Definition: 'J satisfies (PS)', 'J satisfies (PS)_c' (for c in R).

Ex: x*exp(1-x) satisfies (PS)_c iff c is not 0.

Proposition: (1) If there exists a PS-seq for J, and if J satisfies (PS), then J has a critical point.
(2) If there exists a PS-seq for J at level c, and if J satisfies (PS)_c, then J has a critical point (at level c).

Discussion: Tactic to prove existence of critical points (f.ex. weak solutions to PDE):
(1) Prove/study existence of PS-seq's (via topological arguments)
(2) Prove/study convergence of PS-seq's (via compactness arguments).

On (1): Prove/study existence of PS-seq's (via topological arguments, for sublevels):

Definition: Deformation of B (subset of Banach X).
B deformable in A (A subset B subset X, Banach).
Remarks and discussions and examples.

Goal: Study deformability of sublevels J^b in J^a (a < b).
Tool: '(minus) gradient flow'.
(Works in Hilbert space; with enough regularity on J; see below for generalisations).
Thereom (no pf given; as for finite-dim case/ Picard-Lindelöf): Existence of solutions to Cauchy pb/IVP in Hilbert space, with locally Lipschitz cont 'vector field' F. If field F is bounded, then flow defined on all R for all initial values.

Definition: C^{1,1} - functional on Hilbert space.

[BS] pp 145-150.

For examples of existence/non-existence of critical points for functions of finitely (mainly 2) variables, see for example:

I. Rosenholtz and L. Smylie, "The Only Critical Point in Town" Test, Math. Mag. 58, 149-150, 1985.

A. M. Ash and H. Sexton, A Surface with One Local Minimum, Math. Mag. 58, 147-149, 1985.

R. Davies, Solution to Problem 1235, Math. Mag. 61, 59, 1988.

B. Calvert and M. K. Vamanamurthy, Local and Global Extrema for Functions of Several Variables, J. Austral. Math. Soc. 29, 362-368, 1980.

J. Bisgard, The Only Critical Point in Town Test and Its Failure in Two Dimensions, PNW MAA Analysis Flowers Session (Central Washington University), April 20, 2012.

J. Bisgard, Mountain Passes and Saddle Points, SIAM Review, Vol. 57, No. 2. (January 2015), pp. 275-292.

For generalisations of '(minus) gradient flow' to Banach spaces, and for functionals with less regularity ('pseudo-gradients'), see ([BS] 4.1.16):

[AM] A. Ambrosetti and A. Malchiodi, Nonlinear analysis and semilinear elliptic problems CUP (2007), Cambridge (p 120 ('Step 2') and Section 8.1.1 (pp 122-123)).

[K] O. Kavian, Introduction à la théorie des points critiques, Springer (1993) (Chapter 4.3 pp 204-206).

[R] P. H. Rabinowitz, Minimax Methods in Critical Point Theory with Applications to Differential Equations, CBMS Regional Conference Series in Mathematics, vol. 65 (American Mathematical Society, Providence, 1986) (App A, pp 81-85).

[S] M. Struwe, Variational Methods. Applications to Nonlinear Partial Differential Equations and Hamiltonian Systems, 4th ed. (Springer, Berlin, 2008) (Chap II 3, pp 81-87).

Merry Christmas, Happy Hanukkah, & Happy Newyear!

13 January:

Resume:
(1) 'Direct Method': Results on existence of weak solutions to Dirichlet BVP for semilinear elliptic PDE's via minimisation.
Next:
(2) 'Minimax Methods': Motivation/idea/abstract goal: Characterize critical value (of functional J) as 'minimax' over suitable class S of sets ('minimax class'). Example: S family of singletons gives the case of minimiser.
Main example/abstract goal: Mountain Pass Theorem (MPT); formulation and description of geometrical interpretation. (The corresponding Maple-file is here.)

Recall: Definition of PS-sequence (at level c), PS-condition (PS) (at level c: (PS)_c); 1D-example from last: x*exp(1-x) satisfies (PS)_c iff c is not 0.

Tactic to prove existence of critical points (f.ex. weak solutions to PDE):
(1) Prove/study existence of PS-seq's (via topological arguments - f.ex. MPT).
(2) Prove/study convergence of PS-seq's (via compactness arguments).

Existence of PS-seq: Only obstruction is deformation of sublevels. Will be constructed via (minus) gradient flow.

Recall: Definition: C^{1,1} - functional on Hilbert space.

Corollary (of 'Picard-Lindelöf in Hilbert space'): For J a C^{1,1}-functional on Hilbert space H, a unique solution to the minus gradient flow exists. (Problem: Might not be defined for all times).

[BS] pp 145-150; 4.2.2; 4.2.5; 4.3.1; 4.3.2.

20 January:

Remark: J decreases along the solution to the minus gradient flow.

'Deformation Lemma': J a C^{1,1}-functional on Hilbert space H, a < b. Assume there is no PS-level for J in [a,b]. Then J^b is deformable in J^a.
Remarks: I.e. if J^b is _not_ deformable in J^a (can often be 'seen' via change in topology), then there _is_ PS-seq at some level in [a,b].

'Deformation Lemma, Local Version': J a C^{1,1}-functional on Hilbert space H, c in R.
If there is no PS-seq at level c, then, for ε>0 small, J^{c+ε} is deformable in J^{c-ε}, with deformation fixing J^{c-2ε}.

Corollary: J a C^{1,1}-functional on Hilbert space H, c in R.
_Either_: There exists a PS-seq at level c _or_: For ε>0 small, J^{c+ε} is deformable in J^{c-ε}, with deformation fixing J^{c-2ε}.

Definition: A family of subsets (of Hilbert space H) invariant for a deformation η.

Definition: Minimax class Γ. Minimax level c associated to Γ ('Γ works at level c').

Definition: An admissible minimax class Γ for a functional J on a Hilbert space H.
Remarks.

[BS] pp 150-154.

27 January:

Main _abstract_ result of 'Critical Point Theory':
Theorem: J a C^{1,1}-functional on Hilbert space H, Γ admissible class at level c. Then there exists a PS-seq for J at level c. If also J satisfies (PS)_c, then there exists critical point for J at level c.

First consequence:
Theorem: J a C^{1,1}-functional on Hilbert space H, which is bounded from below, then there exists a PS-sequence at level c, with c = inf J. Remarks.

Second (the main) consequence:
Mountain Pass Theoprem (MPT): J a C^{1,1}-functional on Hilbert space H, which has the Mountain Pass Geometry ('two valleys', at 0 and v). Then there exists a PS-seq for J at level c, with c the minimax level of the minimax class of continuous curves connecting 0 and v. If also J satisfies (PS)_c, then there exists critical point for J at level c.

Proposition: Omega open and bounded subset of R^N, q bounded, q>=0, p in (2,2^*) ('superlinear, subcritical'). Then J(u)= (1/2)||u||_q^2 - (1/p)|u|_p^p has the Mountain Pass Geometry.
(Recall: J is not bounded from below. u=0 is critical point of J.)
Remarks.

Major application of MPT:
Theorem: Existence of _a_ non-trivial weak solution to 'superlinear, subcritical'
Nonlinear BVP for (q bounded, q>=0)
- Δu + qu = f(u)
with conditions on (cont.) f:
(f)_6 |f(t)-f(s)|<= C|t-s|(1+|s|+|t|)^{p-2} for some p in (2,2^*).
(f)_7 lim_{t to 0} f(t)/t = 0.
(f)_8 For some M>2, μ>2: f(t)t >= μ F(t) when |t|>=M.
(f)_9 For some t_0 with |t_0|>=M: F(t_0) > 0.

Remarks. (Proof next time).

[BS] pp 154-158; 160-161.

03 Februar:

Remarks on 'Main application of MPT' (see above):
f(u) = |u|^{p-2}u, p in (2,2^*), special case.
(f)_6 is (i) growth condition and (ii) makes f locally Lipschitz.
This allows to prove J is C^{1,1}. (f)_6 _can_ be replaced with 'superlinear & subcritical growth' (which gives J is C^1), _if_ prove stronger version of MPT via 'pseudo-gradients'.

Proof of 'Main application of MPT': Via three (3) lemmas, giving (1) J is C^{1,1} (2) J satsifies (PS)_c for all c (3) J has Mountain Pass geometry.
By MPT gives critical point, which delivers (non-trivial) weak solution.

Statement and proof of three lemmas mentioned above.

[BS] pp 160-165.

End of lectures!


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Letzte Änderung: 04 February 2016 (no more updates).

Thomas Østergaard Sørensen






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