Richard Courant

How to use this course

Richard Courant is a rare case of a mathematician whose institutions are as famous as his theorems. Students meet his name in the Courant–Fischer minimax principle, the nodal domain theorem, the Courant–Friedrichs–Lewy condition of numerical analysis and the book Methods of Mathematical Physics. Many more met mathematics itself through What is Mathematics?. He also built the Mathematical Institute at Göttingen and, after his expulsion by the Nazi regime, the institute at New York University that now bears his name.

Each technical lesson opens with a paragraph for the general reader, then gives precise statements, proof sketches and exercises with folded solutions. The thread running through Part II is Courant’s lifelong method: turn a problem of physics into a problem of minimizing an energy, then use the minimization both to prove existence and to compute.

Timeline

Silesia and Breslau, 1888–1907

Richard Courant was born on 8 January 1888 in Lublinitz, a small town in Upper Silesia, then part of Prussia. He was the eldest son of Siegmund Courant, a Jewish businessman, and Martha Freund. The family moved to Glatz and then, when Richard was nine, to Breslau (now Wrocław). The philosopher Edith Stein was his cousin through his mother’s family.

His father’s businesses failed more than once, and the family’s money was never secure. When the family moved to Berlin in 1905, Richard stayed behind in Breslau, supporting himself by tutoring. He left the König-Wilhelm Gymnasium early, passed the university entrance examination in 1906 as an external candidate, and studied at the University of Breslau.

Breslau was a good school in mathematics, but its best young men were already looking to Göttingen. Otto Toeplitz and Ernst Hellinger, both from Breslau, had gone there to work with David Hilbert. After a semester in Zurich in 1907, Courant followed them. He arrived in Göttingen on 1 November 1907.

Hilbert’s Göttingen, 1907–1914

Göttingen in 1907 was the centre of the mathematical world. Felix Klein ran its institutions, Hilbert set its research agenda, and Hermann Minkowski was about to give space-time its geometry. Courant found his way quickly. In 1908 he became Hilbert’s assistant, preparing lecture notes and working closely with him on analysis.

Hilbert had just rescued Dirichlet’s principle, the idea of finding a harmonic function by minimizing an energy, from the objections of Karl Weierstrass (Lesson 04). Courant’s dissertation of 1910, Über die Anwendung des Dirichletschen Prinzipes auf die Probleme der konformen Abbildung, used the rehabilitated principle to prove existence theorems for conformal maps. The method of the thesis, existence through minimization, guided his work for the next sixty years.

He completed his habilitation in 1912, with an inaugural lecture on existence proofs in mathematics on 23 February. In the summer of that year he married Nelly Neumann, a mathematician from Breslau. Among the young Göttingen mathematicians and physicists of these years were Hermann Weyl and Max Born, his friend from Breslau.

War, Münster and return, 1914–1920

When war broke out in 1914 Courant was drafted into the infantry. He served on the Western Front, fell ill with typhoid, and was seriously wounded on 27 September 1915. Back in uniform after his recovery, he turned his skills to technology. With colleagues he developed an earth telegraph, a device that sent signals through the ground between trenches, where wires were constantly cut by shelling. The project taught him to work with industry and the army, and to raise money, skills he used for the rest of his life.

His first marriage ended in divorce in 1916. Nelly Neumann was murdered by the Nazis in 1942. On 22 January 1919 Courant married Nerina (Nina) Runge, daughter of the Göttingen applied mathematician Carl Runge. They had four children: Ernest, who became a particle physicist; Gertrude, a biologist, who married the mathematician Jürgen Moser; Hans, a physicist; and Leonore (Lori), a violist, whose second husband was the mathematician Peter Lax.

In 1920 Courant accepted a chair at Münster. He stayed only a few months. When Erich Hecke left Göttingen for Hamburg, Klein and Hilbert brought Courant back to succeed him. In the same year he published a paper on the eigenvalues of the differential equations of mathematical physics, which contains the minimax principle of Lesson 05.

Dirichlet’s principle and conformal mapping

For the general reader. A soap film stretched on a wire frame settles into the shape with the least energy. Bernhard Riemann used the same idea to find solutions of the equations of electrostatics: among all candidate functions, take the one with the least energy. Weierstrass showed that a least-energy function need not exist. Hilbert repaired the argument, and Courant made it a working tool, starting with his thesis on conformal maps.

The principle

Let \(\Omega\subset\mathbb R^2\) be a bounded domain and \(g\) a function on \(\partial\Omega\). The Dirichlet integral is

Why the minimizer is harmonic. If \(u\) minimizes and \(\varphi\) vanishes on the boundary, then \(t\mapsto D[u+t\varphi]\) has a minimum at \(t=0\), so

for all such \(\varphi\), hence \(\Delta u=0\). Why a minimizer exists. Take a minimizing sequence \(u_n\). The parallelogram law gives

where \(d\) is the infimum, so \((u_n)\) is Cauchy for the energy norm. Completeness of \(H^1_0\), which Riemann did not have, supplies the limit. Hilbert’s 1900 rescue used a direct compactness argument; the Hilbert-space proof above is the later form that Courant’s school made standard.

Weierstrass’s objection

In 1870 Weierstrass pointed out that a positive functional need not attain its infimum. His example is a one-dimensional variational problem whose infimum is 0 but is approached only by functions that converge to a discontinuous limit (Exercise 4.1). Riemann had assumed that a minimizer exists because the energy is bounded below. Making existence rigorous required both the right function space and lower semicontinuity of the energy.

Conformal mapping

For a simply connected domain \(\Omega\) with a smooth boundary and a point \(z_0\in\Omega\), let \(u\) be harmonic in \(\Omega\) with \(u(z)=\log|z-z_0|\) on \(\partial\Omega\). Then \(G(z)=\log|z-z_0|-u(z)\) vanishes on \(\partial\Omega\), is harmonic except for a logarithmic pole at \(z_0\), and is negative inside. With \(\tilde u\) a harmonic conjugate of \(u\), the function

is single-valued, holomorphic, vanishes only at \(z_0\) and has \(|f|=1\) on \(\partial\Omega\). An argument-principle count shows that \(f\) maps \(\Omega\) one-to-one onto the unit disc. This is the Riemann mapping theorem, obtained from one application of Dirichlet’s principle. Courant’s thesis extended such arguments to multiply connected domains and to mappings onto canonical slit domains, and his 1950 book collected the theory.

Eigenvalues: the minimax principle

For the general reader. A drum can sound many notes, and each note corresponds to a shape of vibration. The notes are the eigenvalues of the Laplacian on the drumhead. Courant showed in 1920 that each note can be found as the answer to a game of minimizing and maximizing energy. The characterization makes it easy to compare drums: shrink the drum and every note rises.

Matrices

Let \(A\) be a real symmetric \(n\times n\) matrix with eigenvalues \(\lambda_1\le\cdots\le\lambda_n\), and let \(R(x)=\frac{x^\mathsf TAx}{x^\mathsf Tx}\) be its Rayleigh quotient.

Proof of the first equality. Let \(e_1,\dots,e_n\) be orthonormal eigenvectors. For \(V=\mathrm{span}(e_1,\dots,e_k)\), \(\max_VR=\lambda_k\). Conversely, any \(k\)-dimensional \(V\) meets the \((n-k+1)\)-dimensional span of \(e_k,\dots,e_n\) in a nonzero vector \(x\), and \(R(x)\ge\lambda_k\). The second equality is proved the same way. Courant’s form, the max–min, is useful because it does not require knowing the earlier eigenvectors.

The Dirichlet Laplacian

For a bounded domain \(\Omega\subset\mathbb R^d\), the problem \(-\Delta u=\lambda u\) in \(\Omega\), \(u=0\) on \(\partial\Omega\), has eigenvalues \(0<\lambda_1<\lambda_2\le\lambda_3\le\cdots\to\infty\).

Extending by zero embeds \(H^1_0(\Omega_1)\) in \(H^1_0(\Omega_2)\), so the minimum for \(\Omega_2\) is taken over a larger family. The Neumann eigenvalues \(\mu_k\) have the same characterization over \(H^1(\Omega)\), which gives \(\mu_k\le\lambda_k\).

Weyl’s law by bracketing

Hermann Weyl proved in 1911 that the counting function \(N(\lambda)=\#\{k:\lambda_k\le\lambda\}\) satisfies

where \(\omega_d\) is the volume of the unit ball; for \(d=2\), \(N(\lambda)\sim\frac{|\Omega|}{4\pi}\lambda\). Courant gave a flexible proof by Dirichlet–Neumann bracketing. Cover \(\Omega\) by small squares. Imposing extra Dirichlet conditions on the inner edges raises eigenvalues, and replacing them by Neumann conditions lowers them, by the minimax principle. For squares the eigenvalues are explicit, and the two bounds agree to leading order as the squares shrink.

Nodal domains

For the general reader. Sprinkle sand on a vibrating plate and it collects along lines that do not move. These nodal lines cut the plate into regions. Courant proved in 1923 that the \(k\)-th mode of vibration cuts the plate into at most \(k\) regions. Higher notes can have finer patterns, but never more pieces than their rank allows.

Let \(u_k\) be an eigenfunction for \(\lambda_k\) on a bounded connected domain \(\Omega\). A nodal domain of \(u_k\) is a connected component of \(\{x\in\Omega:u_k(x)\ne0\}\).

Proof sketch. Suppose \(u=u_k\) had nodal domains \(D_1,\dots,D_m\) with \(m\ge k+1\). Let \(\varphi_j=u\) on \(D_j\) and 0 elsewhere. Each \(\varphi_j\in H^1_0(\Omega)\), and integrating by parts on \(D_j\) gives \(\int|\nabla\varphi_j|^2=\lambda_k\int\varphi_j^2\). The \(\varphi_j\) have disjoint supports. Take a nonzero combination \(\varphi=\sum_{j=1}^kc_j\varphi_j\) orthogonal to \(u_1,\dots,u_{k-1}\); it exists because there are \(k\) coefficients and \(k-1\) conditions. Then the Rayleigh quotient of \(\varphi\) equals \(\lambda_k\), and by the variational characterization \(\varphi\) is itself an eigenfunction for \(\lambda_k\). But \(\varphi\) vanishes on the open set \(D_{k+1}\), contradicting the unique continuation property of solutions of \(\Delta\varphi+\lambda_k\varphi=0\).

In one dimension, Sturm’s oscillation theorem of 1836 says more: the \(k\)-th eigenfunction of a regular Sturm–Liouville problem has exactly \(k-1\) interior zeros, so exactly \(k\) nodal domains. In higher dimensions equality fails for most \(k\). Åke Pleijel showed in 1956 that equality holds for only finitely many \(k\).

Difference equations and the CFL condition

For the general reader. To solve the equations of physics on a computer, one replaces smooth space and time by a grid. In 1928, nearly two decades before electronic computers, Courant, Kurt Friedrichs and Hans Lewy asked when grid solutions approach the true ones. For waves they found a simple rule: in one time step, the computation must be able to “see” at least as far as the wave can travel. Every weather and climate model obeys it.

Domains of dependence

Consider \(u_t+a\,u_x=0\) with \(a>0\). The solution is \(u(x,t)=u_0(x-at)\), so \(u(x,t)\) depends only on the initial value at one point \(x-at\). On a grid \(x_j=j\Delta x\), \(t^n=n\Delta t\), an explicit scheme computes \(U_j^{n+1}\) from \(U_{j-p}^n,\dots,U_{j+q}^n\). After \(n\) steps, \(U_j^n\) depends on initial data in \([x_j-pn\Delta x,\ x_j+qn\Delta x]\), the numerical domain of dependence.

Proof. Suppose the physical point \(x-at\) lies outside the numerical interval. Change the initial data near \(x-at\) only. The true solution at \((x,t)\) changes; the numerical solution does not, for any grid in the refinement. So the numerical solutions cannot converge to the true one for both sets of data.

The upwind scheme

For \(a>0\), \(U_j^{n+1}=U_j^n-\nu\,(U_j^n-U_{j-1}^n)\). If \(0\le\nu\le1\) this is a convex combination \((1-\nu)U_j^n+\nu U_{j-1}^n\), so \(\max_j|U_j^n|\) never grows: the scheme is stable. Consistency plus stability gives convergence, by the Lax equivalence theorem proved at the Courant Institute in 1956. For \(\nu>1\) the CFL condition fails and the scheme diverges.

The elliptic case

The same paper proved that the five-point discrete Laplacian converges to the continuous one for the Dirichlet problem, and linked discrete harmonic functions to random walks: the value at a grid point equals the expected boundary value at the point where a random walk started there first leaves the domain. This is one of the earliest uses of random walks in analysis.

Variational methods and finite elements

For the general reader. Engineers today compute the stresses in bridges, aircraft and teeth by cutting the object into small triangles or tetrahedra and approximating the solution by simple functions on each piece. In a 1943 lecture Courant proposed exactly this, as a way to compute the minimum of an energy. The idea was rediscovered by engineers in the 1950s and named the finite element method.

The setting

The Poisson problem \(-\Delta u=f\) in \(\Omega\), \(u=0\) on \(\partial\Omega\), is the Euler–Lagrange equation of

Walther Ritz (1908) proposed minimizing \(E\) over a finite-dimensional subspace \(V_h\subset H^1_0\). Ritz used global polynomials. Courant’s 1943 paper, written up from his 1941 address to the American Mathematical Society, used the space of continuous functions that are linear on each triangle of a triangulation and vanish on the boundary. Its basis consists of “hat” functions \(\phi_i\), equal to 1 at vertex \(i\) and 0 at all other vertices. He applied it to the torsion problem for a beam.

Writing \(u_h=\sum_iU_i\phi_i\) and minimizing gives the linear system \(KU=F\), with stiffness matrix \(K_{ij}=\int\nabla\phi_i\cdot\nabla\phi_j\) and load vector \(F_i=\int f\phi_i\). Because each hat function overlaps only its neighbours, \(K\) is sparse, symmetric and positive definite.

Proof. Both minimizers satisfy the weak equation \(\int\nabla w\cdot\nabla v=\int fv\) for test functions in their spaces; subtracting gives orthogonality. Then for any \(v_h\), \(\|\nabla(u-v_h)\|^2=\|\nabla(u-u_h)\|^2+\|\nabla(u_h-v_h)\|^2\) by Pythagoras. So \(u_h\) is the orthogonal projection of \(u\) onto \(V_h\) in the energy inner product. With interpolation estimates for piecewise linear functions this gives an error of order \(h\) in the energy norm for smooth \(u\).

Supersonic flow and shock waves

For the general reader. A supersonic aircraft or an explosion pushes the air faster than sound can carry warning ahead. Pressure then jumps almost instantly across a thin layer, a shock wave. During the Second World War Courant and Friedrichs wrote the mathematical account of such flows for military research, and published it in 1948 as Supersonic Flow and Shock Waves. It founded the modern theory of hyperbolic conservation laws.

Characteristics and breaking

The simplest model is Burgers’ equation \(u_t+\big(\tfrac12u^2\big)_x=0\), or more generally \(u_t+f(u)_x=0\). Along curves with \(dx/dt=f'(u)\), the solution is constant. These characteristics are straight lines, and if a faster one starts behind a slower one, they cross.

Indeed \(u_x=\dfrac{u_0'(\xi)}{1+t\,u_0'(\xi)}\) along the characteristic from \(\xi\), which blows up when \(1+tu_0'(\xi)=0\).

Weak solutions and jump conditions

Beyond \(t^*\) one admits discontinuous weak solutions, functions satisfying \(\iint(u\varphi_t+f(u)\varphi_x)\,dx\,dt+\int u_0\varphi(x,0)\,dx=0\) for all smooth compactly supported \(\varphi\).

Integrate the equation over \([a,b]\) containing the shock: \(\frac{d}{dt}\int_a^bu\,dx=f(u(a))-f(u(b))\). Splitting at \(s(t)\) and letting \(a,b\to s\) gives \(s'(u_--u_+)=f(u_-)-f(u_+)\). Weak solutions are not unique; the physical one is selected by an entropy condition, which for convex \(f\) says that characteristics enter the shock from both sides: \(f'(u_+)<s'<f'(u_-)\). Courant and Friedrichs set out these principles for gas dynamics, and Peter Lax, a student of Friedrichs and later Courant’s son-in-law, gave the general theory in the 1950s.

Gas dynamics

For one-dimensional isentropic flow of a polytropic gas with velocity \(u\), sound speed \(c\) and adiabatic exponent \(\gamma\), the quantities \(r_\pm=u\pm\frac{2c}{\gamma-1}\) are Riemann invariants: \(r_\pm\) is constant along characteristics \(dx/dt=u\pm c\). This is how Courant and Friedrichs solved piston problems, rarefaction waves and nozzle flows.

Plateau’s problem and minimal surfaces

For the general reader. Dip a twisted wire loop in soapy water and a film forms, using the least area the wire allows. Proving that such a surface always exists was Plateau’s problem. Jesse Douglas and Tibor Radó solved it around 1930. Courant then showed that Dirichlet’s principle, the tool of his thesis, gives a simpler solution and extends to films of more complicated shape.

From area to energy

Let \(\Gamma\subset\mathbb R^3\) be a Jordan curve. Look for \(x:\bar B\to\mathbb R^3\) on the closed unit disc, mapping \(\partial B\) monotonically onto \(\Gamma\), that minimizes area \(A[x]=\iint|x_u\times x_v|\). Area is invariant under all reparametrizations, which destroys compactness. Douglas and Courant instead minimized the Dirichlet integral \(D[x]=\frac12\iint(|x_u|^2+|x_v|^2)\), which satisfies \(A[x]\le D[x]\) with equality exactly when \(x\) is conformal. A minimizer of \(D\) turns out to be harmonic and conformal, hence a minimal surface.

The key lemma

Proof. In polar coordinates about \(p\), let \(L(\rho)=\int_{C_\rho}|x_\theta|\,d\theta\). By Cauchy–Schwarz and \(|x_\theta|\le\rho|\nabla x|\), \(L(\rho)^2\le2\pi\rho^2\int_{C_\rho}|\nabla x|^2d\theta\). Divide by \(\rho\) and integrate from \(\delta\) to \(\sqrt\delta\):

The lemma gives equicontinuity of the boundary values of a minimizing sequence, once three points of \(\partial B\) are fixed to map to three chosen points of \(\Gamma\). The three-point condition removes the noncompact group of conformal automorphisms of the disc. With equicontinuity, the Arzelà–Ascoli theorem produces a convergent subsequence and a minimizer. Courant extended the method to surfaces of higher genus and to free boundary problems, and presented the theory in his 1950 book.

The Göttingen institute and 1933

Back in Göttingen from 1920, Courant combined research with a talent for organization that his colleagues found remarkable. He edited Springer’s yellow series of monographs, the Grundlehren der mathematischen Wissenschaften, which he had helped launch, and wrote books that became classics. In 1922 he completed Adolf Hurwitz’s lectures on function theory as Hurwitz–Courant. In 1924 he published volume I of Methoden der mathematischen Physik, which carries Hilbert’s name as co-author because it drew on Hilbert’s lectures, but which Courant wrote. Its account of eigenvalue problems, published just before quantum mechanics, became a handbook for the physicists who created it. His calculus lectures appeared in 1927–29.

His students in Göttingen included Kurt Friedrichs, Hans Lewy, Franz Rellich, Herbert Busemann, Otto Neugebauer, Fritz John and Edgar Krahn. Courant’s greatest project was a home for the mathematical sciences. Göttingen’s mathematicians were scattered among rented rooms. With Klein’s backing and funding from the Rockefeller Foundation’s International Education Board, a new Mathematical Institute was built in 1927–29 and dedicated on 2 December 1929. Courant directed it.

1933

Within weeks of the Nazi seizure of power, the civil-service law of 7 April 1933 removed Jewish officials and political opponents. Courant had served at the front, which in principle protected him, but he had been active on the social-democratic left. On 5 May 1933 he was notified that he was placed on leave. Colleagues collected signatures on his behalf, without effect. In the same months Emmy Noether, Weyl, Bernays and many others were driven out, and the Göttingen school was destroyed in a few months. Courant accepted an invitation to Cambridge for the academic year 1933–34, and in 1935 he was formally retired from Göttingen.

New York, death and legacy, 1934–1972

In 1934 Courant accepted a two-year position at New York University, which had almost no research mathematics. In 1936 he became professor and head of the graduate department. He brought Friedrichs from Germany, and later Fritz John and James Stoker, and built a group devoted to analysis and its applications. He became a US citizen in 1940. With Herbert Robbins he wrote What is Mathematics? (1941), which introduced generations of readers to number theory, geometry, topology and calculus. In 1937 the second volume of Methoden appeared.

During the Second World War Courant served on the Applied Mathematics Panel of the Office of Scientific Research and Development, and his group worked on shock waves, underwater explosions and gas dynamics. The postwar result was Supersonic Flow and Shock Waves (1948), and a new institute. NYU created its Institute of Mathematical Sciences in the early 1950s with Courant as director, and the Atomic Energy Commission placed a computing centre with a UNIVAC there. Courant retired as director in 1958. In 1964 the institute was renamed the Courant Institute of Mathematical Sciences, and in 1965 it moved into Warren Weaver Hall.

His honours included election to the American Philosophical Society (1953) and the National Academy of Sciences (1955), the Navy Distinguished Public Service Award (1958), the Grand Cross of Merit with Star of the Federal Republic of Germany (1959) and the Distinguished Service Award of the Mathematical Association of America (1965). In 1965 he also published, with Fritz John, the revised Introduction to Calculus and Analysis. He suffered a stroke on 19 November 1971 and died in New Rochelle, New York, on 27 January 1972, aged 84.

Legacy

• Analysis: the minimax principle, the nodal domain theorem, the Courant–Lebesgue lemma and the direct method of the calculus of variations are standard tools of spectral theory and geometric analysis.
• Computation: the CFL condition and the piecewise linear elements of 1943 are at the root of numerical methods for partial differential equations.
• Books: Courant–Hilbert, Courant–Friedrichs and Courant–Robbins remain in print.
• Institutions: the Göttingen Mathematical Institute and the Courant Institute, whose members have included Friedrichs, John, Stoker, Lax, Moser, Louis Nirenberg and S. R. S. Varadhan.
• Students: his doctoral students included Kurt Friedrichs, Hans Lewy, Franz Rellich, Herbert Busemann, William Feller, Otto Neugebauer, Fritz John, Harold Grad, Joseph Keller, Martin Kruskal and Anneli Lax.

Sources and further reading

Biography

1. J. J. O’Connor and E. F. Robertson, Richard Courant, MacTutor History of Mathematics.
2. Richard Courant, Wikipedia.
3. Courant, Richard, Neue Deutsche Biographie / Deutsche Biographie.
4. New York University Archives, Richard Courant Papers, biographical note.
5. C. Reid, Courant in Göttingen and New York: The Story of an Improbable Mathematician, Springer, 1976.
6. P. D. Lax, Richard Courant, Biographical Memoirs of the National Academy of Sciences.

Mathematics

1. R. Courant, Über die Anwendung des Dirichletschen Prinzipes auf die Probleme der konformen Abbildung, dissertation, Göttingen, 1910.
2. R. Courant, Über die Eigenwerte bei den Differentialgleichungen der mathematischen Physik, Math. Z. 7 (1920).
3. R. Courant and D. Hilbert, Methoden der mathematischen Physik, 2 vols., Springer, 1924 and 1937; English edition Methods of Mathematical Physics, Interscience.
4. R. Courant, K. Friedrichs and H. Lewy, Über die partiellen Differenzengleichungen der mathematischen Physik, Math. Ann. 100 (1928) 32–74; English translation IBM J. Res. Dev. 11 (1967) 215–234.
5. R. Courant, Variational methods for the solution of problems of equilibrium and vibrations, Bull. Amer. Math. Soc. 49 (1943) 1–23.
6. R. Courant and H. Robbins, What is Mathematics?, Oxford University Press, 1941.
7. R. Courant and K. O. Friedrichs, Supersonic Flow and Shock Waves, Interscience, 1948.
8. R. Courant, Dirichlet’s Principle, Conformal Mapping, and Minimal Surfaces, Interscience, 1950.
9. P. D. Lax, Hyperbolic Systems of Conservation Laws and the Mathematical Theory of Shock Waves, SIAM, 1973.