Vorlesung: Semi-linear Elliptic PDEs 2 (SoSe2016)

Vorlesung: Semi-linear Elliptic PDEs 2 (SoSe 2016)

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.

Further literature referred to below:

[E] L. C. Evans, Partial Differential Equations: Second Edition, AMS (Graduate Studies in Mathematics), 2010.

[R] M. Růžička, Nichtlineare Funktionalanalysis, Springer, 2004.

[N] J. Nečas, Direct Methods in the Theory of Elliptic Equations, Springer (Springer Monographs in Mathematics), 2012.

[D] B. Dacorogna, Direct Methods in the Calculus of Variations, Springer (Applied Mathematical Sciences), 2008.

[FL] I. Fonseca and G. Leoni, Modern Methods in the Calculus of Variations: Lp Spaces, Springer (Springer Monographs in Mathematics), 2007.

[G] E. Giusti, Direct Methods in the Calculus of Variations, World Scientific, 2003.

[G1] Giaquinta, Introduction to Regularity Theory for Nonlinear Elliptic Systems, Birkhäuser (ETH Zürich Lectures in Mathematics), 1993.

[G2] Giaquinta, Multiple Integrals in the Calculus of Variations and Nonlinear Elliptic Systems, AMS (Annals of Mathematics Studies 105), 1983.

[GM] M. Giaquinta and L. Martinazzi, An Introduction to the Regularity Theory for Elliptic Systems, Harmonic Maps and Minimal Graphs, Springer (Publications of the Scuola Normale Superiore), 2012.

[BF] A. Bensoussan and J. Frehse, Regularity Results for Nonlinear Elliptic Systems and Applications, Springer (Applied Mathematical Sciences), 2002.

[DM] G. Dinca and J. Mawhin, Brouwer Degree and Applications, preprint, January 17, 2009 (forthcoming book).

[GW] Z. Guo and J. Wei, Global solution branch and Morse index estimates of a semilinear elliptic equation with super-critical exponent, Trans. Amer. Math. Soc. 363 (2011), no. 9, 4777–4799.

[AP] Ambrosetti and Prodi, A Primer on Nonlinear Analysis, CUP (Cambridge Studies in Advanced Mathematics), 1995.

[AA] Ambrosetti and Arcoya, An Introduction to Nonlinear Functional Analysis and Elliptic Problems, Birkhäuser (Progress in Nonlinear Differential Equations and Their Applications), 2011.

[KM] Kriegl and Michor, The Convenient Setting of Global Analysis, AMS (Mathematical Surveys and Monographs), 1997.

[W] M. Willem, Minimax Theorems, Birkhäuser (Progress in Nonlinear Differential Equations and Their Applications), 1996,

[SW] A. Szulkin and T. Weth, The method of Nehari manifold, in Handbook of Nonconvex Analysis and Applications, International Press, pp. 597-632, 2014.

[BC] L. Boccardo and G. Croce, Elliptic Partial Differential Equations, De Gruyter, 2014.

[P] R. Precup, Linear and Semilinear Partial Differential Equations, De Gruyter, 2013.

[L] H. Le Dret, Équations aux dérivées partielles elliptiques non linéaires, Springer, 2013.

[K] O. Kavian, Introduction à la théorie des points critiques, Springer, 1993.

[AZ] J. Appell and P. P. Zabrejko, Nonlinear Superposition Operators, CUP, 2008.

[V] M. M. Vainberg, Variational methods for the study of nonlinear operators, Holden-Day, 1964.

[De] K. Deimling, Nonlinear functional analysis, Dover, 2010.

[AM] A. Ambrosetti and A. Malchiodi, Nonlinear analysis and semilinear elliptic problems, CUP, 2007.

[St] M. Struwe, Variational methods, Springer, 2008.

Here is a longer list of books (to be updated).

12 April:

Introduction, practical Information (see main page).

Chapter 1: Notation, repetition, generalisations.

Recall from last semester:

Dirichlet Boundary Value Problem (BVP) for semilinear elliptic PDE; classical solutions; weak solutions.
Existence by _Variational Approach_: u weak solution iff u critical point for appropriate functional. Growth condition on nonlinearity f that ensures (Fréchet) differentiability of functional.
Example: Linear case (Fredholm theory & Lax-Milgram): Sequence of eigenvalues & eigenfunctions (weak solutions); info on solvability of corresponding inhomogeneous equation (Fredholm alternative).
Minimization: If minimum of functional attained, then have weak solution. (Themes: convexity, coercivity, weak l.s.c.).
Repetition of various conditions on the nonlinearity f which ensures existence of weak solutions (possibly unique). Discussion of graphs of various conditions/functions f and relation to linear case.

19 April:

Minimax methods: Characterise critical values as 'minimax' over suitable class of sets. (Themes: subelevels and their topology, deformation (of sublevels), (minus) gradient flow, Palais-Smale (PS) sequences, PS-condition (admissible) minimax classes).
Main 'concrete' example: Mountain Pass Theorem (MPT).
MPT is consequence of main abstract result of Minimax Theory: If there exists an admissible minimax class working at level c, then there exists PS-sequence at level c.
Application: superlinear/subcritical growth of nonlinearity (plus various technical assumptions) give existence of non-trivial weak solution. Special case: |u|^{p-2}u, p in (2,2^*) (with correct sign!)

End of repetition.

General remarks on 'Direct Method' for minimization of energy functionals depending on vector valued functions; Lagrangian; Euler-Lagrange equations (system of quasilinear PDEs), remarks.
Definition: Null-Lagrangian.
Proposition: If L is (smooth) null-Lagrangian, then E(w) only depends on the value of w on the boundary of the domain.

(see
[E] L. C. Evans, Partial Differential Equations: Second Edition, AMS (Graduate Studies in Mathematics), 2010, pp. 455-466,
and
[R] M. Růžička, Nichtlineare Funktionalanalysis, Springer, 2004, pp. 9-17.)

For more on general boundary value problems in the linear case, see for example

[N] J. Nečas, Direct Methods in the Theory of Elliptic Equations, Springer (Springer Monographs in Mathematics), 2012.

For more on the Direct Method, including in the vectorial case, see for example

[D] B. Dacorogna, Direct Methods in the Calculus of Variations, Springer (Applied Mathematical Sciences), 2008.
[FL] I. Fonseca and G. Leoni, Modern Methods in the Calculus of Variations: Lp Spaces, Springer (Springer Monographs in Mathematics), 2007.
[G] E. Giusti, Direct Methods in the Calculus of Variations, World Scientific, 2003.

The latter also treats regularity theory for elliptic systems. For more on this, see for example

[G1] Giaquinta, Introduction to Regularity Theory for Nonlinear Elliptic Systems, Birkhäuser (ETH Zürich Lectures in Mathematics), 1993.
[G2] Giaquinta, Multiple Integrals in the Calculus of Variations and Nonlinear Elliptic Systems, AMS (Annals of Mathematics Studies 105), 1983.
[GM] M. Giaquinta and L. Martinazzi, An Introduction to the Regularity Theory for Elliptic Systems, Harmonic Maps and Minimal Graphs, Springer (Publications of the Scuola Normale Superiore), 2012.
[BF] A. Bensoussan and J. Frehse, Regularity Results for Nonlinear Elliptic Systems and Applications, Springer (Applied Mathematical Sciences), 2002.

26 April:

Lemma: For w: R^N -> R^N a C^2-function, the divergence of the k'th row of cof(nabla w) is zero, for k=1,...,N.
(Proof: Formula from Cramer's rule, a calculation (!), Schwarz' Theorem on 2nd derivatives, and Linear Algebra.)

Theorem: The determinant L(P) = det(P) is a null-Lagrangian.
Remarks.

Theorem: Brouwer's Fixpoint Theorem.

Consequence of:
Theorem: Boundary of (closed) ball B in R^n is not a (continuous) retract(ion) of B.
(Proof: That boundary of (closed) ball B in R^n is not a _smooth_ (C^2) retract(ion) of B follows using that the determinant is a null-Lagrangian. The continuous case follows by smoothing/regularisation/mollification.)

(see [E] pp. 455-466 and [R] pp. 9-17.)

For more on Brouwer's Fixpoint Theorem, its various proofs, and history, see
[DM] G. Dinca and J. Mawhin, Brouwer Degree and Applications, preprint, January 17, 2009 (forthcoming book) (Chapter 23, pp. 281-301 (!)).

03 May:

Consequence of Brouwer's Fixpoint Theorem:
Theorem: For any continuous map φ from B_R(0) (closed ball in R^N) into R^N, equal to the identity on bondary of B_R(0), there exists x in B_R(0) with φ(x)=0.

Chapter 2: More on minimax: Application of Mountain Pass Theorem (MPT), and the Saddle Point Theorem (SPT).

Recall: Definition of Palais-Smale sequence (PS-seq) and Palais-Smale condion (PS), also at level c in R.

Another application of MPT:
Theorem: For Ω open and bounded subset of R^N, N>=3, q in L^infty(Ω), q>=0, p in (1,2^*), and λ < λ_1(-Δ+q), there exists a non-trivial weak solution to the Dirichlet BVP for - Δu+qu = λu+|u|^{p-2}u.
(Proof: Regularity and PS-condition for corresponding functional J proved last semester; that J has Mountain Pass geometry follows from scaling.)

Another consequence of Main abstract result of Minimax Theory:
Theorem: Saddle Point Theorem (SPT):
H Hilbert space, J C^{1,1} functional on H, E_n n-dim linear subspace of H, V its orthogonal compliment in H. B_R^n closed ball in E_n of radius R and centre 0, dB_R^n its boundary _in_ E_n. Assume there exists R so max of J over dB_R^n is strictly smaller than inf of J over V. Then there exists PS-seq for J at some level c larger than inf of J over V. If additionally J satisfies (PS)_c then there exists critical point of J at level c.

Proof: Use the minimax class Γ of continuous maps of B_R^n into H fixing the boundary dB_R^n.
Need to prove: Γ is admissible. (result then follows from Main abstract result of Minimax Theory.)
That the level at which Γ works is finite/real follows using (consequence of) Brouwer's Fixpoint Theorem.
(Next time: Γ is invariant for all deformation fixing certain sublevels of J.)

[BS] pp. 165-167; 158-159. See also [E] pp. 503-513.

(For some information on the Dirichlet BVP for - Δu+qu = λu+|u|^{p-2}u in the super-critical case p>2^*, see for example
[GW] Z. Guo and J. Wei, Global solution branch and Morse index estimates of a semilinear elliptic equation with super-critical exponent, Trans. Amer. Math. Soc. 363 (2011), no. 9, 4777–4799.
and references therein.)

10 May:

End of proof of Saddle Point Theorem (SPT): The minimax class Γ (see last time) is invariant for all deformation fixing certain sublevels of J; hence, Γ is admissible.

Definition: Asymptotically linear (Dirichlet BV) problems.

Definition: Non-resonant (Dirichlet BV) problems, and problems at resonance/resonant problems.

Application of SPT to non-resonant, asymptotically linear, semi-linear elliptic Dirichlet BVP (!):

Theorem: For Ω open and bounded subset of R^N, N>=3, q in L^infty(Ω), q>=0, λ_k the e. values of -Δ + q, h in L^2, λ not eq λ_k;, and f satsifying:
(f)_10: f in C^1, f bounded, |f'(t)| <= C(1+|t|^{2^*-2}).
There exists a weak solution to the Dirichlet BVP for - Δu+qu = λu+f(u)+h(x).

Proof: Lemma: The corresponding functional J is C^{1,1}. Lemma: J satisfies (PS)_c for all c. (Next time: Lemma: J has the needed 'geometry' to apply SPT).

[BS] pp. 159-160; 167-169.

24 May:

Recall: Application of SPT to non-resonant, asymptotically linear, semi-linear elliptic Dirichlet BVP (see above).

Lemma: J has the needed 'geometry' to apply SPT.

(Ends proof of theorem from last time.)

For another technique to study similar problem, see
[BS] pp. 46-55 (!!).

Next: Application of SPT to resonant, asymptotically linear, semi-linear elliptic Dirichlet BVP:

Landesman-Lazer Condition(s). Discussion and remarks (to be continued next time).

[BS] pp. 170-171; 32-33 (1.7.5); 47 (2.2.5); 171-172.

31 May:

Discussion of Landesman-Lazer Condition(s) (continued).

Theorem: For Ω open and bounded subset of R^N, N>=3, λ_k the e. values of -Δ , h in L^2, λ eq λ_{n+1} (k-fold degenerate eigenvalue), and f satsifying:
(f)_10: f in C^1, f bounded, |f'(t)| <= C(1+|t|^{2^*-2}).
Limits f_l and f_r of f at -infinity and +infinity exist and are different.
If h satsfies the Landesman-Lazer Conditions (given by f), then there exists a weak solution to the Dirichlet BVP for - Δu = λu+f(u)+h(x).

Proof: Weak solutions equal critical points of functional J; existence of such: Application of SPT. Need (1) Regularity of J (2) J satisfies (PS)_c for all c (3) J satisfies 'geometry' in SPT.

(1): 'As usual'.

(2) Lemma: J satisfies (PS)_c for all c.

(3) Lemma: J satisfies 'geometry' in SPT (two parts; proof of hard part next time).

[BS] pp. 172-175; 177-178.

07 June:

End of proof of (see last time):
Lemma: J satisfies 'geometry' in SPT.

Outlook (NO PROOFS; see [BS] Sec. 4.5, pp. 178-186.):

Nonlinear eigenvalue problem: - Δu + q(x)u = λf(u) (Semilinear elliptic PDE with parameter λ).
Question: Existence of one or more weak solutions, and dependence on λ. (Typically, u=0 is solution.)

Example: - Δu + q(x)u = λ|u|^{p-2}u. From earlier: No non-trivial solutions for negative λ. At least one (non-negative) solution for λ positive. One-parameter family (in λ) of solutions (for λ positive) with ||u_λ|| to 0 for λ to infinity and ||u_λ|| to infinity for λ to 0.

[BS] pp. 175-177; 178.

14 June:

Theorem: For Ω open and bounded subset of R^N, N>=3, q in L^infty(Ω), q>=0, f:R -> R with
(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)_11 For some μ>0: f(t)t >= μ F(t) for all t in R.
(i) If also f(t) >= M_1 t^{r-1} for all t in [0,t_1] (some r>2 (!), M_1>0,t_1>0) then:
For all λ>0 there exists non-trivial, non-negative weak solution u_λ to the Dirichlet BVP for
- Δu+qu = λ f(u), with ||u_λ|| to 0 for λ to infinity.
(ii) If also F(T) > 0 for all t > 0 then: For all λ>0 there exists non-trivial, non-negative weak solution u_λ to the Dirichlet BVP for
- Δu+qu = λ f(u), with ||u_λ|| to infinity for λ to 0.

Example: For 2 < p < r < 2^*, let f(t) = t^{p-1}+t^{r-1} for t>0, f(t)=0 otherwise.

Idea of proof: MPT (twice), with well-chosen minimax classes to give proof of asymptotics (see [BS] pp. 179-182 for details).

Another example of possible behaviour:
For 1 < r < 2 < s, let f(t) = t^{s-1} for t in [0,1], t^{r-1} for t >= 1, and 0 for t <= 0.
Proposition: For Ω open and bounded subset of R^N, N>=3, q in L^infty(Ω), q>=0, f:R -> R as above, λ in [0, λ_1(-Δ+q)], the trivial solution u=0 is the only non-trivial, non-negative weak solution to the Dirichlet BVP for - Δu+qu = λ f(u).

Theorem: For Ω open and bounded subset of R^N, N>=3, q in L^infty(Ω), q>=0, f:R -> R as above, there exists λ-tilde so for any λ > λ-tilde, there exists two non-trivial, non-negative weak solutions v_λ (minimum), u_λ (MPT crit pt) to the Dirichlet BVP for - Δu+qu = λ f(u), with ||v_λ|| to infinity , ||u_λ|| to 0 for λ to 0.

Chapter 3: Constrained Minimization.

Goal: Two (more) proofs of:
Theorem: For Ω open and bounded subset of R^N, N>=3, q in L^infty(Ω), q>=0, and p in (2,2^*), there exists non-trivial, non-negative weak solution u to the Dirichlet BVP for - Δu+qu = |u|^{p-2}u.

Remarks: Already seen pf via MPT. Functional not bounded below.

Proof 1: Minimization on Spheres (L^p-sphere). Discussion/heuristics of 'nonlinear FA' approach via constrained minimization (on infinite C^1-manifolds) and Lagrange Multipliers. Here: 'pedestrian' proof. (To be continued.)

[BS] pp. 178-186 (Sec 4.5); 188-189 (Sec 4.7 (!)); 55-59.

(For more information on _nonlinear_ FA ('Ana-II in infinite dimensions' ie., differentiability (higher derivatives), partial derivatives, Taylor's formula, inversion theorems, Implicit Function Theorem, constrained minimisation/Lagrange's Theorem etc etc), see for example:
[AP] Ambrosetti and Prodi, A Primer on Nonlinear Analysis, CUP (Cambridge Studies in Advanced Mathematics), 1995,
[AA] Ambrosetti and Arcoya, An Introduction to Nonlinear Functional Analysis and Elliptic Problems, Birkhäuser (Progress in Nonlinear Differential Equations and Their Applications), 2011.
These also contain information on 'Lyaponov-Schmidt Reduction' (related to bifurcation methods).
For a more comprehensive book, see:
[KM] Kriegl and Michor, The Convenient Setting of Global Analysis, AMS (Mathematical Surveys and Monographs), 1997.
For a longer list of literature, see here.)

21 June:

End of Proof 1: Minimization on Spheres (L^p-sphere).

Proof 2: Minimization on the Nehari manifold:
Definition: Nehari manifold N. Lemma: N is non-empty. Lemma: Inf of J on N is positiv. Lemma: Inf of J on N is attained at some non-negative u (proof: next time).
End of Proof 2: u is a (global) critical point of J.

[BS] pp. 59-63. For the fact that |u| is in H^1 if u is in H^1, see Q3 Pb Sheet 3, PDE2 SoSe2015, or [E] Pb 18 (and 17) p. 310.

(For more on the Nehari manifold/(Nehari) natural constraint, see [AA] p. 150-151 and
[W] M. Willem, Minimax Theorems, Birkhäuser (Progress in Nonlinear Differential Equations and Their Applications), 1996,
and
[SW] A. Szulkin and T. Weth, The method of Nehari manifold, in Handbook of Nonconvex Analysis and Applications, International Press, pp. 597-632, 2014.)

28 June:

End of Proof 2: Minimization on the Nehari manifold:
Lemma: Inf of J on N is attained at some non-negative u.

[BS] pp. 59-63.

Outlook (on Nehari): So have:
Theorem: For Ω open and bounded subset of R^N, N>=3, q in L^infty(Ω), q>=0, and p in (2,2^*), there exists (at least) two distinct weak solutions u_1 (=0), u_2 (non-trivial, non-negative) to the Dirichlet BVP for - Δu+qu = |u|^{p-2}u.

Can prove (using Nehari) perturbation of the above theorem:
Theorem: For Ω open and bounded subset of R^N, N>=3, q in L^infty(Ω), q>=0, and p in (2,2^*), there exists ε>0 so for all h in L^2 with |h|_2 < ε there exists (at least) two distinct weak solutions u_1, u_2 to the Dirichlet BVP for - Δu+qu = |u|^{p-2}u + h(x). See [BS] pp. 63-74.

The method of Nehari can also be extended to non-homogeneous nonlinearities (under some conditions). See [BS] pp. 74-85.

Chapter 4: Fixpoint Methods.

Discussion of (linear) solution operator to the inhomogeneous linear Dirichlet BVP for the Laplacian (or, for the Helmholtz equation). Discussion of how to turn the question of existence of weak solutions of semilinear elliptic equation into fixpoint equation for a (nonlinear) operator on some Banach space.

Nemetskii /superposition operators N_g: N_g is (norm-)continuous from L^α to L^β if g is Caratheodory function, and satsfies |g(x,t)| <= a(x) + b|t|^{α/β} with a in L^β and b>=0.
(See see Q2 Pb Sheet 9, PDE2 SoSe2015, or literature below.) Remarks.

Recall on fixpoint theorems: Brouwer's Fixpt Thm, Banach's Fixpt Thm, Kakutani's counterexample in separable Hilbert space.

For more on Nemetskii operators, see:

[R] pp. 67-69; [AP] pp. 15-22;
[BC] L. Boccardo and G. Croce, Elliptic Partial Differential Equations, De Gruyter, 2014; pp. 13-17,
[P] R. Precup, Linear and Semilinear Partial Differential Equations, De Gruyter, 2013; pp. 208-216,
[L] H. Le Dret, Équations aux dérivées partielles elliptiques non linéaires, Springer, 2013; pp. 55-56 and 61-81,
[K] O. Kavian, Introduction à la théorie des points critiques, Springer, 1993; pp. 60-63,
[AZ] J. Appell and P. P. Zabrejko, Nonlinear Superposition Operators, CUP, 2008,
[V] M. M. Vainberg, Variational methods for the study of nonlinear operators, Holden-Day, 1964; pp. 154-178.

05 July:

Theorem (Schauder): M non-empty convex and compact subset of Banach space X, and invariant for action of continuous (nonlinear) map T. Then T has a fixpoint in M. (No proof; see below).

Theorem (Schauder): M non-empty convex and closed and bounded subset of Banach space X, and invariant for action of compact (nonlinear) map T. Then T has a fixpoint in M. (No proof; see below).

Remarks.

Theorem (Schaefer; special case of Leray-Schauder): For T:X->X compact (nonlinear) map (X Banach), _either_ the set {x in X | x = t T(x) for some t in [0,1) } is unbounded _or_ T has a fixpoint in X. (No proof; see below).

See [S] H. Schaefer, Über die Methode der a priori-Schranken, Math. Ann. 1955, Volume 129.

(For various proofs (by various techniques) of Schauder and Leray-Schauder/Schaefer, see:
[BC] pp. 10-12 (using Brouwer's Fixpoint Thm); [R] pp. 21-28 (using Brouwer's Fixpoint Thm); [AA] p. 38 (using Leray-Schauder degree); [L] pp. 48-54 (using Brouwer's Fixpoint Thm); [De] pp. 60-61.)

Application:
Theorem: For Ω open and bounded subset of R^N, N>=3, f continuous & bounded on R, there exists a weak solution to the Dirichlet BVP for - Δu+qu = f(u).

Remarks: Already know (f satisfies (f)_4). Conditions can be relaxed.

Two proofs: Using the two versions of Schauder, one in L^2, one in H^1_0. (See [L] pp. 56-58; [R] pp. 30-32.)

For other applications of Schauder and Leray-Schauder/Schaefer, see:
[P] pp. 213-217; [BC] pp. 89-90 & 173-175; [AA] pp. 86-87.

For a few applications of _Banach's_ Fixpoint Theorem, see [BC] pp. 76-78 (Dolph's Theorem); [P] pp. 211-213.

See also [E] pp. 540-545.

Chapter 5: Method of lower and upper solutions.

Also called 'suB- and supersolutions', and 'Monotone Iterative Method'.

Definition: u <= λ on dΩ for u in H^1(Ω) and λ in R. sup of u over dΩ for u in H^1(Ω).

08 July:

Definition: sub- and supersolutions (also called lower and upper solutions) for Dirichlet BVP for semilinear uniformly elliptic equation.

Theorem: (Weak) Maximum Principle (weak formulation).
Remarks. Relationship to sub-and superharmonic. (Note: The non-negativity of solutions discussed in 'Outlook' in Chapter 2 (see 07 June and 14 June above) is proved using this Maximum Principle.)

Theorem: For Ω open and bounded subset of R^N, N>=3, A(x) uniformly elliptic, h in L^2, f Caratheodory and increasing in t, with |f(x,t)| <= a(x) + b|t| with a in L^2 and b>=0.
Assume u and v, with u <= v, are sub- and supersolutions of Dirichlet BVP for - div(A(x) Du)= f(x,u) + h(x). Then there exists u_*, u^* weak solutions, with u <= u_* <= u^* <= v, and any other weak solution between u and v is beween u_* and u^*.

Proof: Monotone Iteration Method (via Maximum Principle).
Remarks and discussion.

See [BC] pp. 31-34. See also [E] pp. 545-548.

For more on 'Method of lower and upper solutions' / 'Monotone Iteration Method' (and the Maximum Principle), see [P] pp. 186-190 & 218-220; [BC] pp. 75-77; [L] pp. 99-107 (classical max principle) & 107-109 (weak max principle) & 117-121 (sub- and supersolutions method in Hölder spaces); [K] pp. 37 & 42-44; [AA] pp. 20-21 & 76-82 (abstract set-up).

For literature on the use of 'Method of lower and upper solutions' in the study of ODE, see for example
[C] A. Cabada, An Overview of the Lower and Upper Solutions Method with Nonlinear Boundary Value Conditions, Boundary Value Problems, 2010.

Chapter 6: Outlook.

Other problems and generalisations:
(1) Boundary conditions (inhom Dirichlet, Neumann (homogen and inhom), Mixed bdry cond, Robin, general (nonlinear) 1st order PDE (or ψDE)).
(2) More general inhom term in the linear (and semilinear) eq: h in H^{-1}(Ω). (See [E] p. 320).
(3) More general second order uniformly elliptic PDO (0th order term more general).
(4) p - Laplacian (see [BS] pp. 86-93; from non-Newtonian fluid dynamics).
(5) Nonlinearity via convolution with u (not just function of u, Du). Example: Choquard/Pekar.
(6) Classical solutions directly in Hölder spaces (f.ex. via fixpoint methods).
(7) Minimal surfaces. (See [St] pp. 19-25 and review (by Frank Morgan, The American Mathematical Monthly Vol. 95, No. 6 (Jun. - Jul., 1988), pp. 569-575) of S. Hildebrandt and A. Tromba, Mathematics and Optimal Form, Scientific American Library (!!)).
(8) Hamiltonian systems (existence of periodic solutions/trajectories). (See [St] pp. 60-65).
(9) Existence of closed (periodic) geodesics on (compact) manifolds. (See [Kl] W. Klingenberg, Lectures on Closed Geodesics, Springer, 1978. See also the review).

Themes _not_ touched upon: Inverse & Implicit Function Theorems in Banach spaces (including 'Lyaponov-Schmidt reduction'; see [AA] pp. 23-31); Degree Theory/Leray-Schauder Topological Degree (see [AA] pp. 33-45; [De]); Bifurcation Theory (also in Hölder spaces; see [AA] pp. 61-72, [AP]); Lyusternik-Shnirelman Category (see [AM] pp. 143-156); Krasnoselskii genus (see [AM] pp.157-176); discontinuous nonlinearities (see [AM] pp. 188-198).

For an introduction to problems with 'Loss of Compactness' (for example in R^N), see [BS] Chapter 3: pp. 97-143 (!!).

For many advanced topics, and applications, see [St]. See also the review.

Further (very sketchy and incomplete!) notes to this part can be found here.

End of lectures!

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Letzte Änderung: 18 July 2016 (no more updates).

Thomas Østergaard Sørensen

Teaching