> <\body> > <\itemize> : <\equation*> \(x)=exp-1>\ is a >> hump. <\equation*> \>(x)=>\(x/\). Normalization (=1>) is still missing. : <\equation*> \(\)=>et-1>\t. : <\eqnarray*> \|>||r=|\(n/2)>r.>>|\|>|||n>r.>>>> : <\eqnarray*> v\u>||\v\\u+U>v\u>>|v\u-u\v>||U>v\u-u\v>>>> : <\equation*> g|L|>\|>|>+=1+. In particular , . : <\equation*> \f\f|L|>\|p|>|>\|p|> if <\equation*> >+>+\+>=1. >>: If s\r\t\\> <\equation*> =|s>+|t>, L\L>, then L> and <\equation*> |>\|\>+|1-\>. : Every open cover has finite subcover. Metric space:>sequentially compact. Heine-Borel (finite-dim): >closed and bounded. : compact metric space. C(S)> with sup-norm is compact if is bounded, closed and equicontinuous. : has compact closure. : \B> if continuous and precompact for every bounded . : B> linear, continuous, compact: <\itemize> either has a nontrivial solution or > exists and is bounded. ``Uniqueness and Compactness>Existence''. : H\\>, bounded above and coercive> solvable in for every H>. Proof: Build operator :H\H>> that gives (u)=B[u,g]> (Riesz rep.). Prove 1-1 and onto. The point is: . :\ BR, NR, \L(X,Y)> I)>, I>\<\|\|\>Tx\<\|\|\>\\> (X>) >\<\|\|\>T\<\|\|\>\\>. Read as ``linear+pw bounded>uniformly bounded.'' <\itemize> : <\equation*> A\\u+B\u+C=0, where is symmetric WLOG can be rewritten into one of <\eqnarray*> +u+>||>|-u+>||>|\u+>||>>> : <\equation*> div+1>>=0 : <\equation*> det(Du)=K(x)(1+\|D u\|) open. <\itemize> C(U)>: harmonic, subharmonic u\0>, superharmonic. subharmonic <\eqnarray*> |>|||->u(y)\S>>||>|||->u(y)\B>>>> (implies if harmonic) Proof: \u=\u>, then exploit u=\(x+\n)>. u=\|=1>u=u.> bounded, connected, subharmonic, u>> constant Proof: Consider . By MVI, on any ball in . Thus open. But so is sup}>. {u\sup}>, both open>. C(>)> and subharmonic. Then assumes extrema on the boundary. Proof: SMP or: Suppose U> is max and u\0>. Then and u> negative semidef, contradicting u=tr(Du)\0>. If only u\0>, consider \|x\|>, which is strictly subharmonic. Strong>constant, Weak>extrema on boundary. Uniqueness follows directly from the WMP. 0> (!) harmonic, \\U> connected>C> such that C inf u.> Proof: Pick ,x\U>, apply MVP for large and small circle, respectively, then shrink/expand domain by using 0>, take sup/inf. Use cover of balls to repeat argument as necessary. look for radial symmetry <\equation*> \=C+>log r>|>|>r>|3.>>>>> Constant chosen because it gives the right constant to prove \=\> (use Green's second id on a ball surrounding the signularity). )=\(\|x-\\|)>. (only in 2D) Subharmonic functions bounded above are constant. C(>):> <\equation> u(\)=K(x,\)\u\x+U>u\>K(x,\)-K(x,\)\>u\S. Proof: Integrate on B>>, \0>. Remains valid if replaced by with harmonic . G=\>>, )=0> for \U>. Use in (). To get one, we need to find with on U>. (Use method of images.) For a ball, we get the <\equation*> H(x,\)=-\|\\||\r\|x-\\|> <\equation*> u(\)=H(x,\)f(x)\S>. : harmonic> <\equation*> \|x\|u(x/\|x\|) 0>>. Properties of : <\itemize> )=H(\,x)> )\0> on >H(x,\)=0> for \B(0,r)> and S(0,r)> H(x,\)\S=1> also gives >)>\ Proof: Differentiate under integral (using DCT). Prove continuity onto the boundary by <\equation*> u(\)-f(y)=,r)>H(x,\)(f(x)-f(y))\S Use >->-continuity of and split integral into \> and \>. (Method called .) : C(U)> harmonic>satisifes MVP for every U>. Proof: Construct a harmonic function on with on . satisfies MVP on any subcircle, thus it satisfies the strong maximum principle. Thus . completely represented by absolutely convergent Taylor series. M\0\\: \|\>f(y)\|\\|!|r\|>>>>. Real analytic is completely determined by power series (use similar open-set method on >h(y)=0\\}> as SMP) >Analytic:> Consider +i\)>. Find a region of > where is differentiable. Analyticity estimates can be obtained by the MVP applied to >u>, then coordinatewise Gauÿ, giving <\equation*> \|\>u(x)\|\max\|u\|\sup\|u\|. Then iterate this estimate with \|> radius to get <\equation*> \|\>u(x)\|\\||r>\|>max\|u\|. Uniformly (on compact subsets of ) converging sequences of harmonic functions converge to harmonic functions. Proof: Limit is continuous (because of uniform convergence). Now exchange limits (DCT) in MVP and prove harmonicity. > harmonic, increasing and bounded at a point. Then )> converges uniformly on compact subsets to a harmonic function. Proof: above + Harnack inequality. --a compactness criterion:\ )> bounded, harmonic>>uniformly (on compact subsets) converging subsequence > harmonic limit. Proof: )> is equicontinuous because of the derivative estimates and the assumed uniform bound. Satisfies MVI locally. Perron's method: <\itemize> :={v\C(>), v\BC, v }>. sup S> is harmonic. Proof: <\itemize> > is closed under finite max. (MVI) subharmonic, <\equation*> V(x)=>|,r),>>||.>>>>> S\V\S>, V>. Fix a closed ball, grab sequence \u> at a point >. >\max(v,\,v,min BC)>. Replace these by their harmonic lifting > around >. HCT for a limit . Prove on ball by finding SMP uniqueness of harmonic liftings of in-between (u>) functions. \U>/regular boundary point:> C(>)> subharmonic, , U\{y})\0>. >tangent plane>regular >exterior sphere>barrier, )-K(x,)> >exterior cone>regular At regular boundary points, . Proof:\ <\itemize> Fix \0>. > from >-> with . barrier-\>, where -2 max BC> outside a ball around the boundary point in question. subharmonic by def. Show f(x)> on boundary and interior. Do some funky tricks involving , its Perron function, and the maximum principle to show opposite inequality. The Dirichlet problem is solvable for all continuous BC data iff the domain is regular. <\itemize> w\w=\|\w\|> proves uniqueness in (>)>. <\equation*> I[w]=\|\w\|+w*g\x for the RHS. C(>)> solves PDE+BC>it minimizes over C(>),w=RHS\\}>. Proof: PDE>min: Start from <\equation*> 0=(-\u+g)(u-w), use Gauÿ, Cauchy-Schwarz, \1/2(a+b)>. min>PDE: , for C>>. Differentiate by . <\itemize> : <\equation*> u>(x)=>>K(x,y)\(\y)=>\|x-y\|\(\y) Computable for a sphere with uniform charge density (same as point charge), finite line, disk. >=0\\=0>. Proof: Show \f=0> for any C>> by <\equation*> \\f=\\(K\\f)=(\\K)\\f=0. : Harmonic function with BC 1 on compact set and BC zero at infinity. Perron function on ever-increasing balls--independent of exact domains. F> exists:> Proof (if F\C>): by Poisson's boundary representation formula (with both and u>) <\equation*> p(\)=F>K(x,\)p\S|\>>. u\0> by the max principle (1 on the boundary must be the max value)>positivity. Proof (if not): <\itemize> Approximate through shrinking compact sets with >> boundary (>-mollified indicators of ={dist(x,F)\1/k}>. =\>\\>>. Then consider \\([c,1])\F> and use Sard's Theorem to deduce boundary smoothness for a.e. . Generate > by above theorem. >\p> uniformly on compact subsets (Harnack) Prove (\)\R> by using a F>--use Fubini and the generator of the disk potential. (``') Thus >weak-* convergent subsequence supported on F>. Thus convergene of >\p> away from F>. Uniqueness by uniqueness of potentials of measures. <\itemize> In 2D, Riemann mapping theorem guarantees that point regularity is topological, not geometric. Lebesgue's Thorn: Using level sets of the potential of the measure >\x> on , one may construct exceptional points. <\itemize> <\equation*> cap(F)=\(\)=>F>\p\S. If F\C>, Green's 1st id gives <\equation*> cap(F)=>\\F>\|\p\|. \U> regular > <\equation*> \>\cap(F)F\{\\\|x-y\|\\}(\\(0,1)). Properties of capacity: <\itemize> \F\cap(F)\cap(F)> () \; <\equation*> cap(F)=>\(\x)=>p\(\x)=\|x-y\|\(\y)\(\y)=p\(\y)\cap(F). )> nested sequence with F=F>, then )\cap(F)>. (smooth =1> on >, \\\\\>=cap(F)>) B)\cap(A)+cap(B)>. (>\p+p> by WMP. Then use Gauÿ' trick.) B)+cap(A\B)\cap(A)+cap(B)> >)=cap(S(0,R))=R>. : nested spheres B>. B)=cap(B)> (think of the potentials) (F):supp(\)\F, u>(F)\1}> (Smooth approx > to so that >=1> on F>. Then Gauÿ' trick.) : <\equation*> E[\]=\|x-y\|\(\x)\(\y). Mutual energy: <\equation*> E[\,\]=\|x-y\|\(\x)\(\y). Properties: <\itemize> If \|]\\>, then pos.def. CSU \E[\]> strictly convex \0> finite measure on . <\equation*> G[\]=E[\]-\(F)\-cap(F) Proof: <\itemize> )> bounded below ( compact> bdd.) Infimizing sequences are precompact (i.e. have bounded (F)>) is wlsc (take infimizing sequence )>, use to cut off, \>, \> (MCT), consider -\]>) Minimizer is unique (strict convexity) Minimizer is > (Consider Euler-Lagrange Equation) Evaluate minimum <\equation*> =inf{E[\]:\\0, supp(\)\F,\(F)=1}. Proof: Apply Gauÿ' principle to >, choose . <\itemize> : u+div(\)=0> : =-\\u>. Together: =\u>. Parabolic scaling invariance: \x>, \t>. Use conservation of mass (u=0>) to obtain the ansatz g(r*t)>. Plug in heat equation to get the heat kernel <\equation*> k(x,t)=t)>e/4t>. Use <\equation*> 2a>e>\y\2a>e>=>|a>. and in-boxing the ball to show <\equation*> \>k(x,t)\x\0t\0. f> solves =\u> for f> for 0>. for uniqueness: <\equation*> u(x,t)=g(t)x 0>>uniqueness. r}.> =U\[0,T]>,\ V=>all except top ``lid'',\ V>=lid. C(V)>, u-\u\0>, )\V>: <\equation*> u(x,t)\>u(y,s)|(t-s)>\y\s <\itemize> Exists for heat spheres as well. Equality and (V)> implies u=\u>. Proof: Let RHS=(r)>. (0)=u(x,t)>, <\equation*> \(r)=-C(\u-\u)\*\y\s\0 with \0}=E(\)>. open, bounded, connected, C(>)> and satisfies MVI. Then <\equation*> max>>u\maxV>u. If max attained at V>, then is \ constant in >>. Proof: If max attained in interior, then on heat ball. Then a polygonal path reaches every point on >. > > . Proof: Construct parallel solution by Green's functions. Conclude uniqueness by MVP. <\itemize> Work on a lattice. (subharmonic>assume max in interior>E[x+h\]\M>.) Implies discrete Laplacian has trivial null-space>!> Allows . (by solving for > on the boundary) : \|X,\,X]=E[X\|X]>. : subharmonic>)\|X]\u(X)> (just like discrete SMP) [with > a random walk] may be a stopping time. If > is first passage time to U>, then >)]>. (=BC, harmonic) <\equation*> E[f(x+W>)]=\U>H(x,y)f(y)=\U>>f(y). \; : <\equation*> u(x)=avg(u >) Same formula as above holds for continuous-time. (Central Limit Theorem, path space version of it, \k(x,t/2)>. Cylinder sets. Convergence in weak-* topology. Law of iterated logarithm. Proof of CLT: Convolution of densities becomes multiplication after Fourier transform. Use independence. Done.) =\u> with IC . <\equation*> E(f(x+W))=u(x,t) Implications on boundary regularity: <\itemize> by F-K is the Perron function \U> is regular iff =0)=1> (BM immediately exits .) Littlewood's crocodile Lebesgue's thorn <\itemize> : <\equation*> A(\)=>\\\>(\). : <\equation*> lim\\>)|\>=)\(n/2)>. <\equation*> lim0+>(2\t)et>=(2\t)e>A(\\)=\|U\|. (Weyl>Kac: Integrate by parts, rescale. Proof of Kac: represent Green's function in terms of eigenfunctions somehow.) <\itemize> =cu> : <\equation*> u(x,t)=f(x+c*t)+f(x-c*t)+g(y)\y. . : <\equation*> u(top)+u(bottom)=u(left)+u(right). Good/bad BCs, Inflow/outflow. Domain of dependence. Method of reflection. Odd/even extension. : u\u-c\u=0.> , =g>. Fourier Analysis: (\,t)=(\)cos(c\|\\|t)+(\)sin(c\|\\|t/\|\\|t)=(\)cos(c\|\\|t)+(\)\cos(c\|\\|t)>: <\equation*> u(x,t)=>k(x-y,t)g(y)\y+\>k(x-y,t)f(y)\y Needs to coincide with solution formula. For , >> Observe: <\equation*> M(x,r)=| ->u(y)\S satisfies <\equation*> \M=>''>M=\-\M. Similarly, if solves =u>, then > solves the : <\equation*> (M)-\M=0. In 3D, this reduces the wave equation to (r*M)=\(r*M)>, which we can solve by D'Alembert's formula for all . Then <\equation*> u=lim0>|r>. <\equation*> )>e\y>\S=\|t)|c\|\\|>. Treat 2D equation as 3D equation, independent of third coordinate. 3>>: Assume (0)=0>. Define <\equation*> v(x,t)\k(s,t)u(x,s)\s as a temporal heat kernel average. Oddly, v=\v>. Solve this. Rewrite using spherical means. Change variables as =1/4t> and invert using the Laplace transform <\equation*> h(\)=>e\>h(\)\\. Uniqueness by energy norm. \> open <\itemize> (U)\C>(U)>. \\> iff <\itemize> > fixed compact set : )\F> \:>\|\>\-\>\\|\0>. (U)> <\itemize> Convergence: |\>L>>\\\(U):>|\||>\||>>. Examples: \\(U)>. Aside: \L> for q>. (not for >), (A Borel measure that is finite on compact sets.), > function, Cauchy Principal value. >L|\||>=(-1)\|>>\||>>. Differentiation is continuous. : \|\N>c>(x)\>>, adjoint, fundamental solution: >. (\)\C>(\)> <\equation*> |\,\|>\sup\|x>\>\(x)\|\\\\,\. A polynormed, metrizable space (Use 2\|+\|\\|=k>|,\>|>|1+|\,\|>>>). Complete, too. (Arzelà-Ascoli). Examples: <\itemize> \\> (convergence carries over, too.)\ )\\>, but not \>. <\equation*> |^>(\)=\\(\)=)>>e\x>\(x)\x Basic estimates: <\eqnarray*> |^>(\)|L>|>>|>|(1+\|x\|)\(x)\<\|\|\>>>\C|L|>\\,>>|\>>|^>(\)\<\|\|\>>>>|>|(1+\|x\|)x>\\<\|\|\>>>>>|\>|^>(\)\<\|\|\>>>>|>|(1+\|x\|)\>\\<\|\|\>>>>>||^>\<\|\|\>,\>>|>|(1+\|x\|)x>\>\\<\|\|\>>>\|^>\C>.>>>> >\(x)=\(x/\)>. \>\)=\\>\\>. \(x)=\(x-h)>. \\)=e\>\\>. <\equation*> \(x)=)>>e\>|^>(\)\\=\>|^>=\\|^>, where \(x)=\(-x)>. > is an isomorphism of >, with \>=Id>. Proof: Prove \ \>-Id)e>=0>, then for dilations and translations, linear comb. of which are dense in >. > is 1-1, >> is onto, but >=\\>. > isometry of >, > continuous from > to >, where\ <\equation*> +=1,p\[1,2]. In particular , >. Proof: Show > dense in > (see below), extend >, use Plancherel for >. \C>>.\=1>. (x)\N\(N*x)>. >(\)> is dense in (\)>. (p\\>) Proof: \f-f|L|>\0> holds for step functions. Step functions are dense in (\)>.\ \|L|>\C|>> (Young's) Pick a step function such that |>\\>. Now measure <\equation*> \-f|L|>=\-g\\+g\\-g+g-f|L|>. >(\)> is dense in >. Proof: Smooth cutoff. : f|\g|L|>=|>.> Proof: by Fubini. :L(\)\>(\)>, with >\{h:\\\: h(x)\0 (x\\)}>. Proof: > is dense in >. Well-defined: Take ,\\f\L>, show \-\\\0> in >>.\ Goes to >>: . >: <\equation*> |L|>\C(r,s)|L|>. > is of type )> and . > of type ,s)> and ,s)> <\eqnarray*> >|||r>+|r>>>|>|||s>+|s>>>>> Then > of type for \[0,1]>. <\itemize> >, convergence as in >. \\\\\\>. Examples: > functions, >> not, >>, )f|L|>\\>. A tempered distribution is no worse than a certain derivative coupled with a monomial multiplication. \>>C,N\\\\:>||>\|\\|,\|\\|\N>>\>\|L>|>> (continuity). \L|\||>=\)\\||>> for D>, > is reflection and > a mollifier \L> is a >> function, namely (x)=\\||>>, where f(y)=(y-x)>. Proof: 1. > maps to >. 2. > sequentially continuous. 3. \C> (FD). 4. \C>> (induction). 5. \L|\||>=|\||>> (Riemann sums). > is dense in >. Proof: \\>. Fix \>, \\(\\L)\D\L> in >.\ > is dense in >. (because > is already dense in >.) :\\\> for \\> as by L|\||>\||>>. \\> linear and continuous. \|>> continuous. !>unique, continuous extension of > onto >. :\\\> continuous. \=1/(2\)>. \\n>, >=\((n-\)/2)> <\equation*> \(C>\|x\|>)=C>\|x\|)>. Use this to solve Laplace's equation. <\section> Hyperbolic Equations <\itemize> General constant coefficient problem. )=\+\P(D)+\+P(D)> Solve )u=f> by solving the )u=0>, (0)=0>, u(0)=g> and finding <\equation*> u(x,t)=u\s. Treat remaining ICs by solving standard problems for P,\,\P>, each time adding to the right hand side, which can finally be killed with the above approach. Fourier-transforms to ,\)=0>, with =\>. Initial conditions m-2>(\,0)>, (\,0)>. Representation of the solution: <\eqnarray*> ,t)>||>>t>|P(i\,i\)>\\>>|,\)Z>||>>P(i\,i\)t>|P(i\,i\)>\\=>>et>\\=0,>>>> where > is a path around the roots. Classical solution requires C>. Requires T\C,N:> <\equation*> \|\Z(\,t)\|\C(1+\|\\|). A standard problem is >>a > solution for all \(\)>. It's hyperbolic iff c\\:>,i\)\0> for all > and \-c>. Proof: Estimate around in the above representation for . L>>> entire. <\itemize> +f(u)=0>. Why are they called called conservation laws? <\equation*> |\t>u=u=f(u)=f(b)-f(a)\0. +(u)=0>. Assume , <\equation*> u|\t>=u|\x>\x|\t>+u|\t> Compare shape with <\equation*> 0=uf(u)+u, obtain x/\t=f(u)>. : slap test function onto equation, integrate by parts. : <\equation*> =|> Apply weak solution formula across a jump, consider normal geometrically to obtain speed. : Jump IC. > non-uniqueness of the weak solution for jump up: rarefaction wave or shock with correct speed? <\itemize> Add viscosity to get +(u/2)=\u>. Put as an antiderivative of . Gives Hamilton-Jacobi PDE +U/2=\U>. Now try to rewrite that into a linear equation, by assuming =\(u)>. Yields ODE +C\=0>, solution =exp(-U/2\)>. This gives the heat equation =\\>. <\equation*> u=2\|\>=exp(-G/2\)\y|exp(-G/2\)\y>=-y\|t>\- with /2t+U>. =inf argmin G>, =sup argmin G>. well-defined, increasing, (\)\a(\)>, > left-continuous, > right-continuous, go to \>. Equal except for a countable set of shocks. <\equation*> |t>\liminf\0>u>\limsup\0>u>\|t> \BC> (bounded, continuous)>,t)\BV>. Proof: ,a> are increasing>differences in >. Vanishing viscosity solutions are weak solutions. Proof: Pass to vanishing viscosity under integral using DCT and boundedness. Cole-Hopf solutions produce rarefaction for jump up, shock for jump down. More properties: <\itemize> \0>u>> exists except for a countable set. >\BV> with left and right limits. Proof: is a difference of increasing functions. ,t)\u(x,t)> at jumps.\ ``Characteristics never leave a shock.'' Proof: Travelling waves for Burgers with viscosity only exist for \u>. a shock location: <\eqnarray*> >|||u>=(u+u),>>|=u(x,t)+u(x,t)>||| ->>>u(y)\y,>>|-a)>||>>u(y)\y>>>> The last equation here is a momentum conservation equality. Proof: )=G(a)>. : ,\:\\\> smooth are an e/ef pair for +f(u)=0>>> convex, f=\>. Then (u)+\(u)=0> for perfectly smooth solutions, otherwise (u)+\(u)\0> in the distributional sense, which means <\equation*> >>>\(u)v+\(u)v\x\t\0. for smooth non-negative . By the vanishing viscosity method, we get an entropy solution. Proof: Multiply the viscosity-added c.law by >. Use chain rule on (u>)>. Use convexity of > to show one term involving > non-negative. Multiply by a non-negative smooth function, let \0> to obtain entropy inequality. : is an entropy solution of a c.law if is a weak solution that satisfies the entropy condition for every e/ef pair. <\eqnarray*> |\t>(u>)>||(u>).>>>> Assuming a traveling wave solution of the form <\equation*> u>=v>, we find <\equation*> |\t>(u>)=-u)|6>. >> Entropy solutions , , > cuts of the event cone (given by max. speed >=max \|f\|>. Then for \t> <\equation*> >>\|u-v\|\>>\|u-v\|. Proof: Doubling trick, clever choice of test functions. Implies uniqueness. <\itemize> +H(D u,x)=0>. Example: Curve evolving with normal velocity: +u\|>=0>. Non-Example: Motion by mean curvature =u/(1+u)> (parabolic). Example: Substitute u> in conservation laws. PDE is infinitely-many-particle limit of Hamilton ODE <\eqnarray*> >>||H(p,x)>>|>>||H(p,x),>>>> which coincides with characteristic equation of PDE. Mechanics motivation: <\itemize> Lagrange's Equation <\equation*> |\t>L|\q>=L|\x>. Way to see this: If , then symmetric in , so LHS becomes conserved. (Noether's theorem.) Equivalent to Hamilton's ODE (Proof: (q*p-L(x,q,t))>, where is the solution of L(v)>. Action, given path : <\equation*> S(x)=L(>,x,t)\t Principle of least action: >Lagrange's Equation. Proof: v>, derivative by >, the usual. Generalized momentum: L>. Assumed solvable for . Hamiltonian: q-L=2T-(T-V)=T+V>. More general way of obtaining . Assume (dropping dependencies!) convex, \>L(q)/\|q\|=\>. Then <\equation*> H(p)=L>(p)=sup{p\q-L(q)}. Solved when L>, but in a more general sense.\ Duality: Edge>Corner. Subdifferentials. convex>\>=L>. Proof: Prove convexity and superlinearity of >>. Use symmetry <\equation*> H(p)+L(q)\p\q to prove two sides of the equality >=L>. : is IC <\equation*> u(x,t)=infL(>)\x+g(y),x(0)=y,x(t)=x=mint*L+g(y). Proof: Inf bounded above by straight-line characteristic. Lower bound works by Jensen's inequality. . Proof: Always pick particular solutions, prove both sides of the inequality. defined by Hopf-Lax is Lipschitz if is Lipschitz. Proof: Lipschitzicity for given is immediate (pick good ). Transform problem to comparison with by semigroup property. Temporal estimate is screwy, involves special choices in inf. by Hopf-Lax is differentiable a.e. and satisfies the H-J PDE where it is. Proof: Rademacher's Theorem. Prove +H(D u)\0> for forward in time by taking increments 0>, using inequality with Legendre transform. Lipschitz+Differentiable solution a.e. is not sufficient for uniqueness. (45-degree angle trough vs. 90-degree trough) \\> if <\equation*> f(x+z)-2f(x)+f(x-z)\C\|z\| for some . >> is concave. >``can be forced into concavity by subtracting a parabola.'' >> and bounded second derivatives implies semiconcavity. semiconcave> semiconcave. Clever choice of test locations in Hopf-Lax. \\> > <\equation*> Hp>\\\j\|\\|. If uniformly convex. Then is semiconcave (indep. of initial data) Proof: Taylor, mess about with Hopf-Lax. Now H(p,x)> nonconvex. Use +H(D u,x)=\\u>. Locally uniform convergence follows from Arzelà-Ascoli.\ is a > on \{t=0}>, for each C>(\\(0,\))> <\quote-env> has a local maximum at ,t)>>(x,t)+H(D v(x,t))\0> (and min>>). If is a viscosity solution, then it is a viscosity solution. Proof: Convergence is locally uniform as \0>. Thus for each fixed ball around a local strict maximum in , a local maximum in >-v> exists if > is small enough. There, =u>> and =u>> and u>\-\v>. +H(D v)\0> follows. Generalize to non-strict maxima by adding parabolas. A classical solution of a H-J PDE is a viscosity solution. Proof: Maximum of >derivatives are equal>PDE. > function>: continuous. differentiable at >. Then v\C:v(x)=u(x)>, has a strict local max. viscosity solution> satisfies H-J wherever it is differentiable Proof: Mollify touching function, >> maintains strict max., verify definition of Viscosity solution. (Mollification necessary because test functions are required to be >>.) Lip(C)\Lip(C1+\|p\|)>>uniqueness. Proof: doubling trick again. p\\>. <\itemize> |>=\|\k>>u|p|>>. (\)\{u\\(\):D>u\L(\),\|\\|\k}> Banach space. (\)\cl(\(\),|k,p;\|>)>. W(\)>. \\\> open>u\C>(\):-u|l,p;\|>\0.> Proof: Mollification, throw derivatives onto by integration by parts. W(\)>, > bounded>u\C>(\)\W(\):-u|l,p;\|>\0.> Proof: Exhaust > by \{dist(x,\U)\1/k}>. Consider smooth partition of unity > subordinate to \\\|\>>. \\>\(\u)> s.t. -\u|l,p|>\\2>. Give one more set of wiggle room on each side for mollification. \u\C>> because there's only a finite number of terms for fixed point/set. Then estimate >. Typical idea: Consider <\equation*> f>(x)=lim0>| ->f(y)\y. W(\)>, \\\>. Then <\itemize> There exists a representative on > that is absolutely continuous on a line and whose classical derivative agrees a.e. with the weak one. If the above is true of a function, then W(\)>. Proof: WLOG (Jensen). Consequences: > closed wrt. max, min, abs. value, >. > connected, > constant. <\itemize> <\equation*> osc=supU>\|u(x)-u(y)\|. >\{\|u(x)-u(y)\|\C\|x-y\|>}>. >|>\>)|>+supy>\|u(x)-u(y)\|/\|x-y\|>>. >\D>\C>>. Norm: sum over multi-indices. L(\)>, \\1>, M\0>: <\equation*> | ->\|u(x)->(x)\|\x\M*r>. Then C>(\)> and u\C*M*r>.> >> is the mean over . Proof: a Lebesgue point of , B(z,r)>. Then >->\|\2M*r>.> Iteration via geometric series and Lebesgue-pointy-ness yields <\equation*> \|u(x)->\|\C(n,\)M*r>. For two Lebesgue points, <\eqnarray*> \|u(x)->\|+\|>-u(y)\|>|>|)M*r>.>>>> <\itemize> : C(\)>, p\n>> <\equation*> >|>\C, where <\equation*> >>+=\p>\p. Considering what happens when you scale functions u>(x)\u(\x)>, these exponents are the only ones possible. If we choose , then the best constant comes to light by choosing >, giving the isoperimetric inequality. Proof: Suppose at first. Compact support> <\equation*> u(x)\>>\|D u(x\x,y,x,\,x)\|\y(i=1,\,n). Then <\equation*> \|u(x)\|\\\y. Integrating this gives <\equation*> \|u\|\x\\|D u\|\x\|D u\|\x\y by pulling out an independent part and using generalized Hölder. Then iterate the same trick. To obtain for general , use on >> with suitable >. <\itemize> : \\n> <\equation*> I>(x)=\|x\|-n>\L(\). \f|L|>\C|>.> : > convex, \|\\>, )>, W(\)>. Then <\equation*> | -|>>\|u(x)->>\|\C*d| -|>>\|D u\|. Proof: Use calculus to derive <\equation*> \|u(x)->\|\|n>| -|>>>\y. Then use potential estimate. W(\)>, \\1>. If M\0> with <\equation*> \|D u\|\M*r>, for all \>. Then C>(\)> and u\C*M*r>>. Morrey=Poincaré+Campanato in >. W(\)>, p\\>. Then C(\)> and <\equation*> oscu\r|>. If >, is locally Lipschitz. Proof: Use Jensen )>> on Poincaré's RHS. Then apply Campanato. <\itemize> : <\equation*> [u]\sup| ->\|u->\|\x {[u]\\}>. : (\)(*\L(\))\BMO(\)>. Proof: Poincaré-then-Jensen. For a compact domain, \L>\BMO>. <\itemize> \B>: >continuous, linear, injective map. (\)\L>>> for p\n> (Sobolev inequality) (\)\BMO> for (\)\C> (Morrey) > bounded now. <\itemize> (\)\L(\)> for p\n> and q\p>>. Proof: Hölder-then-Sobolev: <\equation*> |>\>>|>\|\\|>>\|>. (\)\C(|\>)> for p\\>. \B>: The image of every bounded set in > is precompact in >. (precompact: has compact closure) Rellich-Kondrachev: <\itemize> (\)\L(\)> for p\n> and q\p>>. U\C>. Our notes do not.> Proof: <\itemize> Grab a >-bounded sequence >. Mollify it to >> Use an >-derivative trick to show >-u|L|>\\|L|>\0> Interpolation inequality for >: >-u|L|>\>-u|L|\>>-u|L>>|>\0>, also using GNS. For fixed >, >> is bounded and equicontinuous (directly mess with convolution). Use Arzelà-Ascoli and a diagonal argument to finish off. (\)\C(|\>)\L(\)> for p\\>. Proof: Morrey's Inequality, then Arzelà-Ascoli. <\itemize> D u+d*u>. Motivation: Calculus of Variations. : W(\)>, C(\)> <\equation*> B[u,v]\>(D vA*D u+b\D v*u)-(c\D u+d*u)v*\x. : on >, > on \>, i.e. with <\equation*> F(v)\>D v\f-g*v d\x. Assumptions: <\description> >)> \\0: \A\\\\|\\|> >)> L>>, i.e. A)|L>|>\\>, >(|>+|>)+>(|>)\\>. >)>0> weakly, i.e. <\equation*> >d*v-b\D v\x\0 for C(\)>, 0>. >'' on the boundary>: v\(u-v)\0>>\W(\)>. : \>u=inf{k\\:u\k \\}>. is a subsolution>>F(v)>>g+div f>. <\equation*> 0=A*DU+b*\D u+d*u (Not equivalent!) Holds if 0>. 0\B[u,v]\0> for 0> and )>, )>, )>. Then >u\sup\>u>. Proof:\ <\itemize> Use 0> for 0> and )> to establish <\equation*> D vA*D v-(b+c)D u\v\d(u*v)-b\D(u*v)\0. Note that is the new test function in )>. Consequently <\equation*> D vA*D v\(b+c)D u\v. Suppose \>u\k\sup>u>. Set \{u\k}> and achieve a |>\C|>> estimate by using ellipticity, the above and boundedness. Use the Sobolev inequality to get >>|>\\\\|\\|>>|>>, and so \|\0> independently of . Let sup>> to obtain a contradiction. (Note >\\> because W(\)>.) Remarks: <\itemize> Implies uniqueness. No assumptions on boundedness, smoothness or connectedness of >. Implies uniqueness. <\itemize> > bounded, )>, )>, )>. Then !> solution of the generalized Dirichlet problem. <\itemize> Reduce BC to > by subtracting arbitrary function and handling RHS. Prove coercivity estimate <\equation*> B[u,u]\|2>>\|D u\|\x-\\>\|u\|\x. (Uses: )>, )>, \a+b/\>. (For >, Poincaré suffices to show coercivity.) \(W)>> is compact.\ <\equation*> Id=*\\>)|\>\\L)|\>>. >\L-\Id>. (\> has negative eigenvalues already. But they might be pushed upward by the first- and zeroth-order junk. So we might have to make them even more negative to succeed.) >>[u,v]=B[u,v]+\|>>, coercivity is maintained. Lax-Milgram shows existence of inverse >> for the not-so-bad operator >>. Start with , introduce >>, multiply by >> and see what happens. Weak maximum principle provides uniqueness for , so that the Fredholm alternative provides existence when combined with Rellich. <\itemize> Assumptions: <\itemize> )>: )>, )>. )>: L(\)>, L>, n>. )>, . \ , Lipschitz. Then for \\\> we have <\equation*> (\)|>\C(\)|>+(\)|>. Proof: <\itemize> Finite Differences. <\itemize> : )>, )>. W> a subsolution, 0> on \>. Then: <\equation*> sup>u\C|L(\)|>+k, where <\equation*> k=>|>+|>. Proof: <\itemize> : )>, )>. W> a subsolution. Then: <\equation*> supu\CR|L(\)|>+k(R), where <\equation*> k(R)=|\>|>+R|>. : )>, )>, W(\)> a supersolution and 0> in \>. Then <\equation*> R(B(2R))|>\CinfB(y,R)>u+k(R). )>, )>, W(\)> a solution with 0>. Then <\equation*> supu\Cinfu+k(R). )>, )>, )>, > connected, W> a subsolution 0>. If <\equation*> supu=sup>u, then . Proof: Weak Harnack shows >u}> is open. >u}> is relatively closed in >. Therefore >u}=\>. ?> : )>, )>, W> solution of . Then is locally Hölder and <\equation*> oscu\C*R>R>sup)>\|u\|+k if R\R>. Proof: <\itemize> > open, bounded. <\itemize> Idea: solution , smooth variation >, functional . >I(*u+\\)\|=0>=0>. Integrate by parts, > was arbitrary>PDE. \\> deformation, \\n>>, n>[\\]\\>. <\equation*> I[u]=>F(D u(x),u)\x. Looking for \>I[u]>, where =W(\)>. > open, bounded <\equation*> I[u]=>\|D u\|-g u\x. <\itemize> Bounded below: a+b/\>, Sobolev (>\2>), Hölder as |>\>>|>\|\\|>, gives <\equation*> I[u]\c|2>->|2>. Bounded above by |2>+|>>. wlsc because convex. strictly convex (unproven)>uniqueness. : \u\I[u]\liminf\>I[u]>. convex> wlsc in (\)>. Proof: Use representation of convex as limit of increasing sequence }> of piecewise affine functions. Implies F(D u)|k>F(D u)> (weak convergence > linear/affine functions). Then <\equation*> F(D u)| >>>liminf\>F(D u)=liminf\>I[u] and MCT. : >>lim g)\>>lim F(g)>. : Weak form obtained from )=I[u+\v]>, where and looking at (0)=0.> <\equation*> -div(F(D u))+F(D u,u)=0. Also (0)\0>. : (s)=>. =1> a.e.. >(x)=\\(x)\(x\\/\)>. <\equation*> v>|\x>(x)\\(x)\(x\\/\)\\(x)\. Consider (0)\0>>DF\\0> pops out. >(wlsc>convexity). Proof: ``>'': shown above. ``>'': > cube grid on >, C>>. <\eqnarray*> (x)>||>v(2(x-cell center))+z\x.>>|(x)>||(x-cell center))+z.>>>> \z\x>, \D u>. Then <\equation*> F(z)|wlsc>liminf\>>F(D u)=>F(z+D v) Thus has a minimum at the straight line, and for )=I[u+\v]>, (0)=0>, (0)\0>, convexity follows as above. 2>, =W\{u=g}\>>. p\\>. C>>, c\|A\|-c>. <\equation*> I[u]=>F(D u(x))\x <\itemize> Sawtooth calculation yields <\equation*> (\\\)DF(P)(\\\) >\\))> convex in . : >A\\n>>, C>([0,1],\)>: <\equation*> F(A)\>F(A+D v) If C(1+\|A\|)>, then QC> wlsc. <\itemize> ``>'': Subdivide domain into cubes, <\equation*> >F(D u)\>F(>)|QC>>F (D u)+errors. Use measure theory to keep concentrations (Dirac bumps?) of or > away from cube boundaries. Mop up the error terms. ``>'': cubes calculation above. : is a convex function of minors of . Convex>PC>QC>R1C (converse false). Proof of PC>QC: PC>wlsc (use convex>wlsc argument for each minor). wlsc>QC. C(1+\|A\|)>. Proof: Exploit growth estimate above, and QC>R1C. Use \\))>, which is convex>locally Lip>locally (0)\|\max\|f\|>. <\itemize> is a if E-L <\equation*> div(D F(D u))=\x\>F(D u))=0 holds for every C>. null Langrangian. Then <\equation*> u= \\\I[u]=I[]. Proof: )\I[\u+(1-\)u]>. (\)=0> by E-L. <\itemize> )=det(Ai,\j>)>. =*(cof A)>. Acof A=det A\Id> \>det(A)=(cof A)> is a null Lagrangian, i.e. <\itemize> = Plug and chug if 0>, otherwise add Id>. \u> in >, p\\>. >)\det(D u)> in >. (Morrey/Reshetnyak) <\itemize> Reduce dimension of problem by one by reducing to ``does the cofactor matrix converge''? Use cof(D u)u>. Morrey (p>!) implies uniform boundedness in >, then use A-A to extract uniformly converging subsequence, settling the deal for the leftover besides the cofactor matrix. (also holds for if )\0>--no proof.) : . There is no continuous map >\\>> with on B>. Proof: Suppose there is a retract . By comparison with and identity on the boundary, <\equation*> det(D w)=\|B\|. OTOH, =1\(D w)w=0\det(D w)=0>. Lose smoothness requirement by continuously extending by Id, mollifying and using then. : >\>> continuous. x\>:u(x)=x>. Proof: Assume no fixed point. \B> is the point on B> hit by the ray from to . is a retract because hits B> in if \B>. is continuous. W> <\equation*> deg(u)=| ->det(D u). Definable for continuous functions by approximation. Is an integer. open, \G\(0,\)> space-time. <\itemize> : <\equation*> D u/D t=(\\u-\p)+f, )> is the material derivative /D t=\\+u\\\>. =0>>Euler equation. \0>>may as well assume =1>. : \+div(\u)=0>. Assume /D t=0>>\u=0>. : Take div of NSE. : \u+\p=\\u>. : ideal (=0>), steady flow \u+\p=0>>(u/2+p)=0> >/2+p=const> (still need conservation of mass \u=0>) : =curl u> <\eqnarray*> \+\\(u\\u)>||\,>>|\u>||>|\u>||.>>>> In 2D, \(u*\\u)> becomes \u>. : closure {\\:\\C>(\)}>. >> (note closed!) is . =P\P>> Example: Divergence-free field from sem. 1 final: (continuous) boundary-normal field matters, (discontinuous) tangential field does not. : <\itemize> Take C>(,\)> div-free, dot NSE with it, i. by parts second term, popping the derivative onto -product, pull apart, one term is zero, a\\p=-(div a)p=0> gives <\eqnarray*> ->\a*\u+\a\(u\u)+\a\u\x\t>||>|>\\\u>||(\\C>(\))>>>> (where B=tr(AB)>) |V|>closure{a\C>(,\),\\a=0}> <\equation*> \>\|a\|+\|\a\|\x\t. Space for ICs: \P\Lclosure {C>}> to replicate on G>. \P>>. u\V>:\ <\itemize> (W1), (W2) : )-u|L(G)|>\0> as 0>, :\ <\equation*> |\t>|>=2u|L|>. Equivalently for 0>, <\equation*> \|u(x,t)\|+\|\u(x,s)\|\x\s\\|u(x)\|\x. <\initial> <\collection> <\references> <\collection> > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > <\auxiliary> <\collection> <\associate|toc> |math-font-series||1General Stuff> |.>>>>|> |math-font-series||2Equations> |.>>>>|> |math-font-series||3Laplace's Equation> |.>>>>|> |3.1Energy Methods |.>>>>|> > |3.2Potentials |.>>>>|> > |3.3Lebesgue's Thorn |.>>>>|> > |3.4Capacity |.>>>>|> > |math-font-series||4Heat Equation> |.>>>>|> |4.1Difference Schemes and Probabilistic Interpretation |.>>>>|> > |4.2Hearing the shape of a drum |.>>>>|> > |math-font-series||5Wave equation> |.>>>>|> |math-font-series||6Distributions/Fourier Transform> |.>>>>|> |6.1Tempered Distributions |.>>>>|> > |math-font-series||7Hyperbolic Equations> |.>>>>|> |math-font-series||8Conservation Laws> |.>>>>|> |math-font-series||9Hamilton-Jacobi Equations> |.>>>>|> |math-font-series||10Sobolev Spaces> |.>>>>|> |10.1Campanato |.>>>>|> > |10.2Sobolev |.>>>>|> > |10.3Poincaré and Morrey |.>>>>|> > |10.4BMO |.>>>>|> > |10.5Imbeddings |.>>>>|> > |math-font-series||11Scalar Elliptic Equations> |.>>>>|> |11.1Existence Theory |.>>>>|> > |11.2Regularity |.>>>>|> > |11.3Harnack Inequality Stuff |.>>>>|> > |math-font-series||12Calculus of Variations> |.>>>>|> |12.1Quasiconvexity |.>>>>|> > |12.2Null Lagrangians, Determinants |.>>>>|> > |math-font-series||13Navier-Stokes Equations> |.>>>>|>