A wall structure (21) for a burner (19), with a first wall (40), which has elevations (41) and which defines a first supply chamber (22), and with a second wall (43), which has recesses (44) cooperating with end sections (42) of the elevations (41) and which defines a reaction chamber (20). The first wall (40) has first openings (34), which connect the first supply chamber (23) to the reaction chamber (20). The second wall (43) has second openings (37), which connect a second supply chamber (24) located between the walls (40, 43) to the reaction chamber (20). The first openings (34) are arranged in first rows (35) and the second openings (37) in second rows (38). The mixture formation or homogenization of the gases in the reaction chamber (20) can be improved if at least one second opening (37) is arranged within the first row (35) between two openings (34).